Monads are rather like minds. (Actually, this understates the matter, since all minds are monads.) They are the basic objects out of which everything else is composed. Since they are basic, they are indivisible; since anything with extension is divisible, they are not extended.
There are three main varieties of monads:
Monads are windowless (7); this means that they do not interact with anything outside themselves. They are created by God, but after that they just unfold according to their natures.
2. Two great principles
the principle of contradiction (31) and the principle of sufficient reason (32; presupposed e.g. in 9)
We know necessary truths by deducing them using the principle of contradiction. The idea is that a truth is necessary if and only if its negation logically implies a contradiction. So every necessary truth can be proved by reductio ad absurdum: assume that it's not true. Show that this assumption leads to a contradiction. Conclude that your starting assumption was false.
We know contingent truths by way of the principle of sufficient reason. The idea here seems to be that we can deduce contingent truths, but only if we start by making some contingent assumptions (in particular, the assumption that God has created the best of all possible worlds).
3. Arguments for God’s existence
4. Best of all possible worlds
53-58 argues that the sufficient reason for the existence of the contingent truths is that of all the possible ways the world could have been, this is the best (as good a tradeoff as possible of variety and order). This is of course the view parodied in Candide, in which Voltaire has Dr. Pangloss espouse it. (This whole metaphor of possible worlds has proven to be very useful.)
This provides Leibniz with his response to the problem of evil: the evil things in the world are there because God could not have removed them without creating a less perfect world (so any alternative world would be even worse). One problem with this response is that the sense in which this world is "best" for Leibniz doesn't seem to have anything to do with pain and suffering. For Leibniz, God could easily have created a world in which humans do not suffer. Why didn't he? Apparently because it would have harmed the balance between simple natural laws and rich phenomena: either the world would have had more complex laws, or else the phenomena would have been more limited. To which one is inclined to reply that it seems a lot more important to have minimal suffering than to have simple natural laws!
5. Argument against materialism
Leibniz offers a famous argument against materialism in section 17. (This is the "mill" argument.)
What is Leibniz's own view? It often seems to be a substance dualism much like Descartes's: a person is a nonphysical mind, but has an associated physical body. However, he sometimes seems to be, not a dualist, but an idealist: nonphysical monads are the only things that really exist, and material bodies exist only in the ideas of those monads.
6. Pre-established harmony
Since monads do not interact, we need some account of apparent interactions between things. The answer is that monads have been created in such a way that their natural unfolding is related one to another. In particular, mind and body are related by way of preestablished harmony (78-80). Minds “act according to the laws of final causes,” while bodies “act according to the laws of efficient causes,” but there is a harmony between the two of these. #80 suggests that this view is forced by Cartesian assumptions together with conservation laws. [Do physical objects interact? Or do they just appear to? --if physical objects are really just ideas in the minds of monads, then do we say (a) they don’t interact, or (b) they do interact; interaction just is defined in terms of certain sorts of things physical objects do (rather like Putnam on brains in vats, or for that matter like Berkeley on speaking with the vulgar).
7. Relational view of space and time
One argument: if space were an infinitely large container, God could have no
sufficient reason to put objects in one place rather than another, since
different locations would make absolutely no difference to the spatial relations
of the relevant objects. [If space were a finite box, then he might have such
reasons as: putting the Earth (or the sun, or whatever) in the exact center. But
if it’s infinite, then there is no center.]