In mathematics, a set is understood as a collection of distinct elements. The number of times and the order in which the elements of a set occur do not matter.
Sets are usually notated with curly braces to indicate what elements the set has. For example, $\{1,2,3\}$ is the set which contains the elements $1$, $2$, and $3$ and which contains no other elements. Note that both $\{1,1,2,3\}$ and $\{3,2,1\}$ are the same set as $\{1,1,2,3\}$, even though they are written differently.
The following are some basic properties of and notations for sets:
- We write $s \in S$ if $s$ is an element of the set $S$ or, equivalently, that $s$ is in $S$ or $S$ contains $s$.
- Two sets are equal (i.e. are the same set) exact when they have the same elements.
- We can take unions of sets - the union $S_1 \cup S_2$ of two sets $S_1$ and $S_2$ is the (unique) set whose elements are exactly those in $S_1$ or $S_2$ (or both). More generally, we can take unions of any number of sets (even infinitely many sets!).
- We can take intersections of sets - the intersection $S_1 \cap S_2$ of two sets $S_1$ and $S_2$ is the (unique) set whose elements are exactly those in $S_1$
- We can take the difference of sets - the difference $S_1 \setminus S_2$ of the set $S_1$ by the set $S_2$ is the (unique) set whose elements are those in $S_1$ but not in $S_2$.
- There is the empty set $\emptyset$. It is the unique set that contains no elements whatsoever.
- For an object $a$, there is the singleton set $\{a\}$ of the element $a$. It is the unique set containing $a$ as a element and which contains no other elements.
For an axiomatic treatmeant of sets, see the Wikipedia pages on Set theory and Zermelo-Fraenkel set theory. Unless you have a good reason to study axiomatic set theory (e.g. you want to study set theory, logic, or category theory to great depth), I strongly recommend that you do not look into it.