Usually when we refer to a factor we mean a "non-trivial factor," which would then exclude this last example, and exclude considering 1 as a factor of anything. More generally, any element that is a factor or everything else is called a "unit."
A factor is something that divides into something else. Examples include:

• 7 is a factor of 91,
• (x-1) is a factor of $x^2-2x+1,$ and
• in a group G, the identity element, e, is a factor of everything.
If a number N can be written as a (usually non-trivial) product, so that N=pq, then p and q are called factors of N.

By analogy, other mathematical objects: polynomials, matrices etc. can have factors.

The fundamental theorem of arithmetic says that the prime factors of a number are, up to sign and reordering, unique.