group
A group is the mathematical capture of symmetry and reversible action. Think of all the ways you can rotate a square so it looks unchanged, or all the moves of a Rubik's cube: you can combine two moves into one, doing nothing is always an option, and every move can be undone. Any system with these features is a group, no matter what its elements actually are.
Formally, a group is a set G together with one binary operation * that obeys four rules: closure (a * b is always in G), associativity ((a * b) * c = a * (b * c)), an identity element e with e * a = a * e = a for all a, and inverses (each a has some b with a * b = b * a = e). That is the entire definition — four short conditions, and an astonishing amount of mathematics follows from them.
Groups are everywhere: the integers under addition, nonzero fractions under multiplication, symmetries of shapes, shuffles of a deck. The power of the abstract definition is that a theorem proven about “any group” instantly applies to all of them at once. Note that a group needs only one operation; the operation need not be commutative, and when it is, we give the group the special name abelian.
The integers under addition form a group: closed (m + n is an integer), associative, identity 0, and the inverse of n is -n since n + (-n) = 0. The same integers under multiplication are NOT a group, because 2 has no integer multiplicative inverse.
(ℤ, +) is a group; (ℤ, ×) is not.
A common slip is to require commutativity. Groups do NOT have to be commutative — symmetries of a square and shuffles of a deck are genuinely non-commutative. Commutativity is an extra optional property (abelian), not part of the four axioms.