Hobbes

Hobbes's ideal of knowledge is deductive (he was much impressed with Euclid's Elements), and although he wrote on many subjects, his philosophical system is to be found in The Elements of Philosophy, divided in 3 parts, De Corpore (On Body), De Homine (On Man), and De Cive (On Citizen), organized, at least ideally, so that the second follows from the first an the third from the second. However, his most famous work is Leviathan, which ranks among the greatest works on political philosophy ever written. Here we look mainly at the first part of that work.

1.
All thoughts come from sensation, which is but motion produced by a reaction of the heart/brain to the pressure produced on it by objects affecting the organs of sense.
The motion of sensation being impeded, sense "decays" and becomes imagination; depending on its characteristics, it is memory, simple or compounded fancy, dream, understanding. Much memory is experience, which engenders prudence. Understanding is especially important and is imagination aroused by voluntary signs; it's common to human and beasts, although a type of it, arising in connection with speech, is restricted to humans.
NOTES:
Hb's adopts naturalism, materialism and empiricism. Contrast this with D's views.

2.
When unregulated, thoughts succeed each other in the imagination according to associative laws (the value of a Roman penny example in ch. 3). However, it's possible regulate one's train of thoughts by subordinating them to a desire we want to satisfy, so that

from an imagined thing, we imagine the means (the causes) to bring it about. This is common to humans and beasts
from an imagined thing, we imagine what (effects) it can bring about. Beasts don't engage in this because they aren't as curious as we are.

Since all thoughts are from sense, we have no thought of the infinite but as something the bounds of which we cannot imagine. So, we have no conception of God, and the name of God is used only to honor the divine being, not to excite a conception of it. However, we know that God exists as a first eternal cause (Chs.11, 12)
NOTE: contrast with D's account in Med. III.

3.
Speech is used to transfer mental discourse into verbal for two ends:

to communicate with each other
to register and generalize the conclusions of our reasoning. This is done by using thoughts generally through the "apt" imposing of general words or phrases (e.g., "triangle", "having the internal angles be equal to two right angles"). Ideas become general by selective attention: in the proof of triangle's property, the idea used is of a particular triangle, but only those features of it which are general are used. This is underlined by the use of general words. Ch. 4.

When two names (phrases) are connected, they make an affirmation (e.g., man is a living creature). If the extension of "living creature" includes that of "man", then the affirmation is true; otherwise it's false.
First truths arise from the conventional imposition of names: "man is animal" is true because we impose the names "man" and "Animal" on the same thing (De Corpore, 3,8)
NOTE: the idea here is that we favor certain conceptual frameworks among the possible ones in order to acquire power (over nature and over others). This is done by imposing names. Notice, however, that a successful imposition is not wholly arbitrary: it must be "apt" (certain frameworks work better than others).

4.
Reasoning is a type of reckoning involving two types of operation, addition and subtraction:

adding the names of the parts to obtain the name of the whole, e.g., from "quadrilateral", "equilateral", and "rectangular", one gets "square"
subtracting the name of a part from those of the whole to get the name of the remainder, e.g., from "man" ("rational animated body"), one gets "body" by subtracting "rational" and "animated".

NOTE: Although there's such thing as right reasoning, we don't have it naturally. Hence, when disputes arise about it, we need to set up a judge whose reason will count a right reason (Ch. 5).

5.
Philosophy (Science) is knowledge, by reasoning, of effects from the thoughts of their causes (synthesis) and of possible causes from their effects (analysis).
NOTE: Hb here belongs to a long tradition which had become highly developed with the Paduan Aristotelians (e.g., Zabarella). The main idea was to decompose a complex phenomenon into its components (analysis) and then compose it again as their effect (synthesis).
Science,

in contrast with prudence, is the result of industry, apt imposition of names, and good ratiocination.
Is conditional, i.e., hypothetical-deductive.
has power as its end.

6.
All voluntary motions start in the imagination as endeavors (minuscule beginnings of motions). From the notion of endeavor, Hb constructs 4 basic passions:

desire or love is an endeavor toward the object causing it
aversion or hate is an endeavor away from the object causing it
Pleasure is the appearance of what's good, i.e., of what we desire;
pain the appearance of what's evil, i.e., of what we hate.

All the other passions are constructed out of these four (ex. glorying is pleasure at one's perceived power)
NOTES:

Nothing is good or evil per se, but only in relation to someone's desire or aversion. Since there's no summum bonum, happiness is the continual progress of desire from one object to another, depending on one's inclinations.
NOTE: this empasizes the acquisitive nature of humans
In society, the sovereign decides what's good and evil.

Deliberation is a sequence of appetites/aversions, hopes/fears, etc. concerning some future state of affairs; will is the last appetite in deliberation (i.e. the one we act upon). Deliberation is common to humans and beasts.
NOTES:

So, reason is motivationally inert and has only instrumental value, as for Hume.
contrast this to D's views on the will

A man's power is the capacity to obtain, by himself or by the help of others, some future apparent good
A man's worth is the market price for the use of his power; as such, it varies.

7.
Chs. 13-14 of Leviathan contain Hb's famous account of the state of nature, the state in which people without a sovereign live:

Humans are by nature equal in physical and mental powers because even the weakest can kill the strongest by secret machination or confederacy with others.
From equality of ability, arises equality of hope in satisfying one's desires.
Given scarcity, equal hope generates competition, and ultimately enmity among those who want the same thing but cannot have it.
This produces generalized diffidence. In such a situation:

anticipation (augmenting one's power by mastering others) seems most reasonable even for peaceful men, especially because

some others enjoy glory
in order to maintain one's present power, one must expand it.

nothing is unjust or wrong because these presuppose a common law, and hence the common power which is, by hypothesis, missing

This amounts to a state of war of everyone against everyone, i.e. a situation in which everyone is disposed to fight against everyone else.
The result is a miserable life without security, culture, commodious living: a life which is "solitary, poore, nasty, brutish, and short."
Since everyone has the basic right of nature to use one's own power as one sees fit to preserve one's life, and the state of nature is one of war, in the state of nature one has a right to everything, including other people's bodies (presumably, because one may decide that anything is needed for one's preservation).

NOTES:

Hb adopts a form of radical individualism:

by assuming that certain basic human characteristics (including self-consciousness, language, and self-interest) are independent of societal links. This is why, in resolving society into its constituents, he finds individuals, and not parents, or children, or husbands and wives.
by claiming that humans do not seek society by nature per se, but in order to profit from it; in other words, society for Hb has only instrumental value.

Hb's idea of a natural equality breaks with the Aristotelian and Medieval tradition of a state of natural inequality among humans (natural masters/natural slaves, men/women, enlightened/uneligthened). Moreover, the fact that he infers a lack of natural political subordination from equality seems to indicate that he conceived of the former as a subordination to other people, not to the law.
Some, e.g., Pufendorf, have argued that from equality in the state of nature, not a state of war, but one of peace would follow because only fools would quarrel with others of equal strength. However, Hb might reply that there are many fools and that scarcity may push people to fight anyway
Hb's position on the ontological status of the state of nature is unclear. Although he denies that it ever existed all over the world at the same time, he also claim that

American "savages" live in such a state
we do live such a state as soon as authority is absent or even sufficiently weakened (locking of doors, chests, etc.) (ch. 13)
the various countries in the world are, in effect, in a state of nature (Why this last case is relevant is unclear)

8.
However, 1) fear of death, 2) desire for better living, and 3) hope to obtain it, incline people to seek peace, and Hb shows how by discussing rights, natural law, and contracts.
Right
A right is the liberty do use, or avoid using, one's power. To lay down one's right to X is to devest oneself of the liberty of hindering another's right to X, i.e. to have an obligation not to hinder the present owner of the right from exercising it; hindering the right's owner is injustice.
NOTES:

Since the right of nature is for one's survival, one cannot renounce the right to self-defence (this includes accusing oneself without assurance of pardon).
Transferring a right involves transferring the means to exercise it.

Contract
A contract is the mutual transfer of rights among two or more people. A contract which is asymmetrical in time is a covenant . A covenant of mutual trust is one in which the parties promise to perform in the future.
NOTES:

Covenants extorted by fear are obligatory (in the state of nature).
No covenants with beasts
A covenant of mutual trust in the state of nature is, upon reasonable suspicion of non-performance, void.

Laws of Nature
A Law of Nature is a general rule discovered by reason, by which one is forbidden, by commission or omission, to do what's destructive of one's life. There are several such laws, but the most important are the first 3:

One ought to seek peace as far as one has hope of obtaining it; and when one cannot obtain it, one must seek and use all advantages of war.
One must be willing to, when others do so, to lay down one's right to all things and be contented with as much right as one would allow to others.
One must perform one's covenants made (non-performance constitutes injustice).

Problem: "the Foole" believes that it's rational to break a covenant if it's advantageous to do so. But it's never advantageous to break a covenant because:

in the state of nature, knowledge of his injustice prevents him from getting needed allies;
in society, he cannot be sure of not getting caught.

NOTES:

The notion of natural law is old, going back at least to the Stoics and playing an important role in the Middle Ages. It amounts to the idea that there are certain universally moral rules, grounded in reason or in God, in the light of which individual and collective actions ought to be judged. But Hb interprets the notion through his view that reason is subservient to desire, and that what we desire most of all is well-being and self-preservation. So, Hb constructs an ethics based on self-interest
The Laws of Nature always oblige in foro interno; but in foro externo they oblige only when others observe them as well.

9.
In order to exit the state of nature, everyone enters into a covenant with everyone else to transfer all of one's transferable rights to a sovereign (chosen by the majority) who uses their powers as he sees fit to ensure peace and common defense of the subjects. By so doing, the subjects become the authors of the sovereign's actions.
The sovereign (which can be a man, woman, an assembly etc.) has not covenanted with anyone because:

he cannot covenant with the whole people because in the state of nature they're not united as a people.
he hasn't covenanted with each separately because if subject A claims the sovereign has breached the covenant, and subject B denies it, A and B will come to blows because there's no judge to decide.

So, the sovereign has remained, in effect, in the state of nature. As Hb makes clear,

subjects cannot change the form of government
the sovereign's actions cannot be justly accused by the subjects because they are the authors of the sovereign's actions.
the sovereign alone is judge of

what's necessary for peace and defense (this includes control on speech and religion, so that the sovereign is also the head of the church)
what property goes to whom
who the ministers, judges and consellors are.
what counts as good and evil

Hence, the sovereign enjoys absolute power because

his rights are indivisible, since all of them together are necessary for peace and protection (e.g., no point in being the defender of the state without the capacity to impose taxe to maintain an army).
since power can be limited only by greater power, there must be a greatest power in the commonwealth, i.e., an absolute power, i.e. a sovereign . In other words, a state is a closed authority system with a person(s) as a final arbiter (De Cive)

NOTES:

The use of the idea of social contract is old, going back to the Sophists and the Epicureans. However, it was rejected by Plato and Aristotle and played a secondary role in the Middle Ages. It's helpful to distinguish two variants:

A pact of association is a contract among people in the state of nature to submit themselves to a sovereign; it's not a pact between the people an the sovereign (this is how, for Hb, we can abandon the state of nature).
A pact of submission is one between the people and the sovereign in which the people submit to the sovereign for protection.

For Hb, although life under an absolute sovereign can be bad, it's infinitely better than one in the state of nature (Ch 19, 20). However, many, e.g., Lawson, Locke (2T 93; 228), and Rousseau (SC I, 2; I, 4) have questioned this, in part because they have a more benign view of the state of nature. Moreover, although it's in the sovereign's interest not to oppress the subjects too much, sovereigns can be as blind and stupid as everyone else.
A subject S has the right to sue the sovereign if the sovereign grounds his claims against S in a precedent law. (Remember, however, that magistrate are appointed by the sovereign).
Since the end of obedience is protection, a subject is absolved from obedience when the sovereign:

cannot protect him anymore

they

has become subject of another
has banished him

10.
Liberty is absence of external impediments, as power is absence of internal impediments. A man who has been given curare has the liberty, but not the power, to move; a man who's in chain has the power, but not the liberty, to move. Hence, liberty is compatible with determinism.
NOTE: contrast with Descartes.
For Hb it's also compatible with fear, as when a man throws his goods overboard for fear of drowning or acts for fear of the law.
In the political realm, the liberty of subjects has 2 sources:

the silence of the law (Hb, rather optimistically, seems to think that it would involve choosing a job, residing where one wants, doing business, educating one's children, etc.)
the inalienability of self-preservation rights. Hence, no subject is bound to hurt himself or not resist harm, or kill another, or to serve in the military (if he substitutes a soldier in his place). Moreover, "men of feminine courage" may without injustice (but dishonorably) run away from battle, unless they have voluntarily joined.