How to define logic...bleh
The shortest, simplest way that I can come up with reads: A system for evaluating the truth value of statements based on a comparison to the known truth value of other statements.
The important part here is the 'known truth value' part, because when you start you don't have any statements with a known value, you have to make up some things and assume them to be true or not true as a place to start. In logic, philosophy, and geometry class we called them axioms, not the the distinction between geometric, philosophical and logical proofs is meaningful (the former two being a special set of the latter).
In my joyous undergraduate logic class (which was philosophy 110 for the curious), we start with simple axioms like "A=A" and "if A=B and B=C then A=C".
Axioms are important because where you start has a great impact on where you end up. If you want an example read up on
Non-Euclidian geometry. Now, I wouldn't go so far as to say certain kinds of logic (a "kind of logic" being defined by it's particular set of axioms) are right or wrong, but certain sets of axioms are more or less useful for certain things.
Many times different axioms are the root cause of an argument, logical inconsistency or the assignment of different truth values to the same statement, if you will (If only people would realize this more often...). The trouble in any kind of debate setting (particularly one where folks are evaluating the existence of god/s) is getting people to see their own axioms as assumptions and recognizing that questioning of axioms is a very fruitful approach to debate and resolving the issue at hand, if it can be resolved at all.