In mathematics, a term used by professors (and students soon thereafter) to signify insignificance:

"At least one zero of f(x)=x^3-x is trivial."

In matrices, there is a trivial vector space x to the equation Ax=0 for any given matrix A. Sometimes this space amounts to x=0.

Some examples of "nontrivial" items:

The solution set of vectors x in the equation Ax=0 for x != 0.

The number of primes less than or equal to some given number N.

The complete factorization of any particular large number (e.g. (10^174)-7).