What your books and teachers have failed to explain to you is that you're not supposed to know. Since a definition of a term can only be fully understood by someone who understands the terms used in the definition, it's impossible to define everything. So something must be left undefined. Obviously, we would like to leave as few things as possible undefined, so it's pretty cool that mathematicians have found that it's sufficient for (I think) all known applications of math to leave exactly two things undefined: what a set is, and what it means for a set to be a member of a set.
You probably already understand statements like "2 is a member of the set whose members are 1,2, and 3" and "pi is a member of the set of real numbers". If you do, you already understand the two undefined concepts almost as well as a mathematician. The next step for someone learning about sets would be to learn in what ways you are allowed to construct new sets from the ones you already have.
Hmmm!!! Thanks, I think. Interesting because, if I remember rightly, they are now using the term "set" in the "new" math - early elementary and upward. I remember explaining it to a group of fourth-graders as like a set of dishes. They know what a set of dishes is. New Math? They also use "input" and "output" for "plus" and "minus". But, first came "add to" and "take away" which make much more sense to a six-year-old. Easy steps first?