If an element $n$ of the integers or more generally of a ring, e.g. a ring of polynomials, can be written as $n=ab$ with $a,b$ in the same ring, then $a,b$ are factors or divisors of $n$. If one of the factors is a unit (a divisor of 1), the factorization is called trivial. Finding a non-trivial factorization (if one exists) can be a daunting task and is the aim of many algorithms. A good factorization method might even render some strong encrpytion methods obsolete.
Having such a factorization can be very helpful, e.g. to simplify tasks: Finding the roots of $x^5-3x^2-5x^2+15$ may be difficult without knowing the factorization $(x^2-3)(x^3-5)$.