Level 1
Definition (first course in algebra): a monoid is a set $M$ equipped with a special element $e$ and a multiplication $m$, such that $e$ is the multiplicative identity, and the multiplication is associative.The natural numbers, with $+$ and $0$, makes a monoid.
Another example is the string monoid: $M$ is all possible English strings made from the 26 letters, $e$ is the empty string, and $m$ is just putting two strings together.
