Informally, noncommutative means "order matters".
More formally, if is some binary operation on a set, and and are elements of that set, then noncommutative means that doesn't necessarily equal .
Most common operations, such as addition and multiplication of numbers, are commutative. For example, , and .
Examples of noncommutative operations
Composition of functions
If and are functions, then usually, . This can also be written .
For example, suppose and . Then , and . Unless , will not be the same as .
If and are both matrices, then usually, . For example:
Symmetries of a regular n-gon
The symmetries of a regular n-gon form a noncommutative group called a dihedral group.