set
A set is simply a bunch of things gathered together and treated as a single object. Picture a grocery bag: it holds an apple, a banana, and a carton of milk, and you can talk about "the bag" as one thing while still asking what's inside. The things inside are its members, or elements — and a set cares only about which things belong, not the order you list them or how many times you say them. {apple, banana} and {banana, apple} are the very same set.
Two questions unlock almost everything: Is this thing in the set? (membership) and How many things are in it? (size). From these humble beginnings, mathematicians build everything else — numbers, functions, shapes, even infinity itself can be defined in the language of sets. That's why sets are called the foundation of modern math: it's the bedrock that the whole cathedral stands on.
One surprise trips people up: a set's members must be distinct, so repeats simply don't count — {1, 1, 2} is just {1, 2}. And a set can even be empty, holding nothing at all, like an empty bag that's still a perfectly good bag. The empty set is real, useful, and there's exactly one of it.
Set A holds 1, 2, 3. The symbol ∈ means "is a member of," ∉ means "is not," and |A| is its size.
The idea was forged by Georg Cantor in the 1870s, who startled the math world by proving that some infinities are bigger than others. His "set theory" was so controversial it was once attacked as a disease — today it's the agreed-upon language in which nearly all of mathematics is written.