What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about definitions. Is there something I might be missing? How is this kind of thought handled in logic?
"Paradox" is not a formally defined term. Many modern authors use "paradox" for all sorts of surprising or unexpected results; you can see examples by searching for "paradox" on Google News.
A more substantial use of the word "paradox" refers to a result that shows that a particular naive intuition is not sound. For example, the Banach-Tarski paradox shows that is it not possible to have a measure of volume for arbitrary subsets of Euclidean space, if that measure satisfies certain basic properties such as invariance under rigid motions and finite additivity. This goes against a certain naive thought that every subset of Euclidean space must have a well-defined volume (it might be 0 for strangely defined sets of points, but at least it should be defined, this intuition would say).
The classical paradoxes of logic (also called "antinomies") are somewhat different because they show that our intuition about logic itself is not valid. In particular, these show that the naive ways we talk about "truth" and "sets" in natural language lead to contradictions. An example is Curry's paradox, which shows that the naive way we prove "if/then" statements in normal mathematics can lead to false results when combined with self-referential sentences (even when there is no negation in the sentences).
The thing that makes the classical paradoxes more genuinely paradoxical is that it is hard to see where the problem comes from, or any straightforward way to resolve the issue. Consider the sentence of Curry's paradox
If this sentence is true, then 0 = 1
We can prove this sentence in the usual way: assume the hypothesis "this sentence is true" and prove that the conclusion must follow. That is how we prove many other implications in mathematics. But then, because the sentence is true, its hypothesis is true, so its conclusion must also be true: 0=1. It is very difficult to find a hidden assumption in this argument as with the Banach-Tarski "paradox".
One resolution in mathematics is to take refuge in formal logic, where the self-reference of the quoted sentence is impossible. But that does not resolve the issue that the sentence seems to be perfectly clear English, and yet applying the usual methods to it leads to a contradiction.
Two thoughts to add to Carl Mummert's fine answer. He writes that a serious use of the work of "paradox"
refers to a result that shows that a particular naive intuition is not sound.
Perhaps it is slightly better, as indeed his own examples show, to say that the interesting paradoxes are cases where we have a bunch of naive intuitions, which the paradoxical reasoning shows can't all be true together, even though taken separately the intuitions continue to look quite compelling. This is why the interesting paradoxes (like the Liar Paradox, to take another example) can be so recalcitrant: they seem to show that we have to give up some pre-theoretic intuition, but it can be quite unobvious which one is the best candidate for revision. So we have to start exploring the costs and benefits of different revisions, and in many cases, the complications ramify.
You might be interested, for example, to look at http://plato.stanford.edu/entries/liar-paradox/ to see a discussion of the ramifications of the Liar Paradox; or for another example you could try http://plato.stanford.edu/entries/sorites-paradox/
The same truly excellent on-line encyclopaedia (generally very reliable on logical matters) also has a nice article about the role of various paradoxes in spurring on the development of modern logic: http://plato.stanford.edu/entries/paradoxes-contemporary-logic/
You may enjoy W.V.O. Quine's essay "The ways of paradox", which tries to answer many of the same questions. Quine suggests early on:
May we say in general, then, that a paradox is just any conclusion that at first sounds absurd but that has an argument to sustain it? In the end I think this account stands up pretty well.
This accords well with Carl Mummert's comments elsewhere in this thread that “modern authors use "paradox" for all sorts of surprising or unexpected results” and “‘paradox’ refers to a result that shows that a particular naive intuition is not sound.”.
Note that this definition leaves open the possibilities that the sustaining argument could be correct or incorrect. In the first case, Quine calls the paradox "veridical", in the second case, "falsidical". (“Typical falsidical paradoxes are the comic misproofs that $2=1$.”)
It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about definitions.
Put that way, it all sounds very simple. The problem is that the properties can seem very simple, and the misunderstandings can be very deep, and it can be hard to understand how the thing can be mistaken on the one hand and yet so seemingly simple on the other hand. Your word "just" waves away the difficulties. Quine discusses Grelling's paradox in this context. We say that adjectives can be true of things; for example, the adjective "red" is true of red things, or "polysyllabic" is true of polysyllabic things, and of polysyllabic words in particular. Grelling asks us to consider the adjective "heterological", which means "not true of itself". The adjective "red" is heterological, because it is a word, and words have no colors, so the adjective "red" is not red. But "polysyllabic" is a polysyllabic word and so is not heterological. You can guess the next step: is "heterological" heterological?
On your account, this is "just" a misunderstanding about the properties some object can have. What is the misunderstanding there? Is it a misunderstanding when certain adjectives can be said to be true of things? (This is Quine's view.) If so, what is the misunderstanding? If there is something wrong with "heterological", what is it, exactly, and if you disqualify it it, what prevents you from also disqualifying "polysyllabic" or "red"?
Yet so faithfully does the principle [of an adjective being true of the things it describes] reflect what we mean in calling adjectives true of things that we cannot abandon it without abjuring the very expression ‘true of’ as pernicious nonsense.
His essay ends:
Of all the ways of paradoxes, perhaps the quaintest is their capacity on occasion to turn out to be so very much less frivolous than they look.