Math definition: A Cauchy sequence is some sequence of numbers, call it an, such that for every E > 0, there is a positive integer N such that if j and k are greater than N, then |aj-ak| < E.
In plain English: Say you have an infinite sequence of numbers. Normally this sequence would be defined by a mathematical expression, such as an = 1/n = {1, 0.5, 0.333..., 0.25, 0.2, 0.166..., etc.}.
Now pick some positive number E, as small as you like. 0.1, 0.01, 0.001, whatever. (Just don't be a smart aleck and try dividing by infinity.)
In order for this sequence to be a Cauchy sequence (or just "be Cauchy", as the math teachers inevitably end up saying), there must be some point in your number sequence such that any two numbers after that point differ by less than E. If E is very small, N may have to be very large, but as long as it exists, your sequence is Cauchy.
For example: Let's say we use the sequence an = 1/n and E=0.1. Then a good value for N would be 10, since a10 = 1/10 = 0.1 and any two numbers in the sequence after that must be between 0.1 and zero--and therefore their difference will be between 0.1 and zero, too. If E=0.01, N=100, and so on and so forth. No matter how small E gets, the sequence will (eventually) squeeze under it.
So what?: A Cauchy sequence is technically different from a convergent sequence in that you don't have to know what number it converges to (which can be handy if that number is irrational--see xriso's writeup below). However, it can be proven that where the real numbers are concerned, all Cauchy sequences are convergent sequences and vice versa. This means that Cauchy sequences can be used to determine whether a real-number function is continuous and makes them an essential piece of calculus and real analysis proofs.