Unfortunately I'm not versed in philosophical logic, so let's start with some "different categories of logic".
Scope
The first relevant category of logic is scope, what is the logic talking about. Propositional logic talks about nonparametric propositions. A typical rule (modus ponens) allows you to derive a proposition B given that you know A and that A implies B.
Predicate logic talks about parametric propositions, for example "X is a poster". We can derive "There's X such that X is a poster" from "Tim is a poster", whereas in propositional logic "Tim is a poster" is monolithic and cannot be broken up to "Tim" and "X is a poster".
Predicate logic is also known as first-order logic. If we quantify over predicates then we get Second order logic. This allows us to use concepts such as "There's a predicate P such that P(X) iff X=Tim".
The scope is enlarged in a different direction by Modal logic, where we can talk about events (propositions) eventually happening, happening until some other event happens, and so on.
Rules
The rules of logic people are usually taught constitute Classical logic. In classical logic you're allowed to use proof by contradiction. However, some mathematicians (well, logicians) dislike such rules since they're non-constructive (this is like set-theorists who consider what happens when you're not allowed to use AC, the axiom of choice).
By modifying (restricting) the rules, we arrive at Intuitionistic logic (or Constructive mathematics), in which every proposition that we prove has a constructive proof from the givens. In particular, you can't use a proof of the sort "if X, then Y; if not X, then also Y" unless you can decide whether X is in fact true. The resulting world is nice - for example all functions are continuous. On the other hand, not every theorem you can prove in classical logic is true in the constructive sense, although surprisingly many are (for example, the fundamental theorem of algebra is still provable).
In a different direction, in Infinitary logic the proof is infinite and there are logical rules with infinitely many premises. For example, you could have a rule deducing "P(n) for every natural n" from the (countably) many propositions P(0), P(1), ... In general these logics are not well-behaved, but if you choose the parameters correctly, they are (look for $L_{\omega_1,\omega}$).
Goals
Different areas of logic have different goals. For example, in Structural Proof Theory one goal is to understand how simple a proof system can be made, and what are the consequences. Using this syntactic approach, you can show that certain statements are not provable in certain systems by converting them into a very simple form, bounding the resulting size of the simple proof, and show that it is too short to prove the given statement.
A complementary direction is Model Theory, which is all about what a given set of axioms describes. You can prove for example that a set of first-order axioms cannot characterize the size of the universe (if it's infinite). The model-theoretic approach to showing that some theorem is unprovable is to exhibit a model of the axioms in which the theorem isn't true.
In computer science, people are interested in Proof Complexity, which studies how long it takes to prove statements if you're only allowed to use basic means. The holy grail is proving a statement which is even stronger than $P \neq NP$. A different sub-field constructs systems in which every predicate proven to exist is efficiently computable.
Categorical logic
Unfortunately I don't know anything about categorical logic. Apparently there are some kinds of categories (in the sense of category theory) which describe "universes of set theory" along with their associated logic, which is often constructive. These things are called Topoi (singular Topos).
Philosophers' logic
Philosophers are interested in different aspects of logic. Unfortunately, I don't know much about these, so I'll only give some sporadic examples.
There is some interest in nice, succinct systems of logic whose axioms are "as natural as possible". In mathematical logic, by and large people aren't interested in the exact form of the axioms but in what they describe and what can be proven about the system using them.
Other interest is in what the allowable rules of logic should be. I described classical logic and intuitionistic logic. There are many systems in between (and even systems weaker than the latter), which might be natural given some philosophical stance. Most mathematicians don't even bother with intuitionistic logic, although some logicians (and even people from other branches) are intrigued by it.
There's also some interest in making sense of paradoxes - given a statement which is paradoxical (for example, the Liar paradox), what should its truth value be? The mathematical solution is to avoid self-reference in some systematic way, since mathematicians are more interested (in this case) in the well-being of the system rather than these "monstrous examples".