What is the definition of a set? From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition.
My question is what the definition of a set is? 
I have noticed that many other definitions start with a set and then something. A group is a set with an operation, an equivalence relation is a set, a function can be considered a set, even the natural numbers can be defined as sets of other sets containing the empty set.
I understand that there is a whole area of mathematics (and philosophy?) that deals with set theory. I have looked at a book about this and I understand next to nothing.
From what little I can get, it seems a sets are "anything" that satisfies the axioms of set theory. It isn't enough to just say that a set is any collection of elements because of various paradoxes. So is it, for example, a right definition to say that a set is anything that satisfies the ZFC list of axioms?

One of the reasons mathematics is so useful is that it is applicable to so many different fields.  And the reason for that is that its logical structure starts with "undefined terms" that can then be given definitions appropriate to the field of application.  "Set" is one of the most basic "undefined terms".  Rather than defining it a-priori, we develop theories based on generic "properties" and allow those applying it to a particular field (another field of mathematics, or physics or chemistry, etc.) to use a definition appropriate to that field (as long as it has those "properties" of course).
