Set : The things correctly named by some noun-like expression. As Cantor defined it: "By a 'set' we mean any gathering into a whole ... of distinct perceptual or mental objecs" or "A set is a many which allows itself to be thought of as one".
Set theory was first conceived of by Georg Cantor, in the second half of the 19th Century, to solve problems and puzzles he faced when dealing with infinities in mathematics.
Set theory is currently the standard foundation of mathematics, especially in the form of the Zermelo-Fraenkel set theory, often abbreviated as ZF. For this there are three main reasons:
(1) Set theory is a lingua franca of mathematics, that allows one to define many mathematical concepts clearly and precisely, and to conduct many mathematical arguments clearly and conclusively, while it is not difficult and indeed, as my definition above suggests, intuitively based, in part at least, on how one uses nouns and common names.
(2) Much of standard mathematics - though not all of it - can be developed on the basis of definitions only presuming the axioms of ZF. This constitutes at least theoretically and notationally a great unification and clarification.
Furthermore, formally speaking set theory only needs standard first-order logic and the primitive concept "is an element of", usually written as "e".
(3) Set theory arose from the attempt to make the reasoning with infinities in mathematics, especially in the calculus, that turns around the concepts of infinite series like: 1, 1/2, 1/4, 1/8, .., 1/2n .. and many others, precise and axiomatic, and succeeds in doing so, at the price of or with the benefit of introducing an infinity of infinities of sets. (See powerset.)
The terms "set", "class" and "collection" are often used as synonyms, though it is noteworthy that one of the ways of avoiding paradoxes in set theory, that was introduced by Von Neumann, is to distinguish sets and classes, or also classes and proper classes as follows: What can be an element is always a set, but what cannot be an element while still having elements is a class or proper class.
This neat distinction has the great merit of dissolving Russell's paradox.
There are many introductions to set theory, of which "Naive Set Theory" by Paul Halmos is one of the best, because it is clear, fast-paced and not fussy about things that don't really matter in a first introduction. Also it is a fairly thin book, so that one doesn't need much time to decide whether this is something for one or not.
A good introduction to and exposition of the many difficulties involved in the concept of the infinite is A.W. Moore's "The Infinite".