Integer factorisation is the problem of finding a
non-trivial factor of a given number. A factor of n is
a number that divides n, and non-trivial means neither
1 nor n.
For example, a non-trivial factor of 11111 is 41, whereas
trivial factors are 1, -1, 11111 and -11111.
If n is prime then it has no non-trivial factors. There
are techniques for identifying non-primes that do not
explicitly exhibit a factor, so the question of finding
a factor is interesting.
The RSA public key cryptosystem uses numbers that are hard
to factor, and if a way could be found to factor numbers
quickly then that would effectively break it.