Quote by mathwonk
yes the purpose is to give a way to test whether a sequence is convergent without finding the limit.

I always thought of them as a clever idea to define convergence even if we didn't have anything to converge to. For example in (0,1) the sequence {1/n} fails to converge ... but it "should" be a convergent sequence. It's the space that's deficient, not the sequence itself.
The concept of a Cauchy sequence formalizes that intuition. Then we can say that a space is complete if all the Cauchy sequences converge  that is, if all the sequences that "should" converge, do converge.
So to me they're an important conceptual step in the process of understanding limits and completeness.