So, you might ask "What this definition is useful for?"
Cauchy sequences are a way of constructing the reals out of the rationals.
Now, let's forget all we know about the real numbers, so that the best things we can work with are the rationals.
Numbers like 5/4, -2, 0.0006, 4367582828365978387934.
Say we want to solve the equation x^2 = 2. Of course, one answer is x = the square root of 2, but it is not a rational number.
We have a pretty good idea of where it should be: something higher than 1, but less than 2. We can delve a little deeper, to restrict it between 1.4 and 1.5. Deeper still: 1.41 and 1.42. If we try hard enough, we can make its square as close to 2 as we want, but not exactly at 2. Let's construct a sequence that keeps adding one more decimal figure of the "square root of two": {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}. Note that every member of this sequence is a rational number.
If we try to find out which rational number this sequence is converging to, then we're out of luck because the sequence does not converge to any such number. This is where Cauchy sequences come to the rescue, because this sequence is in fact Cauchy.
The result: We have a sequence of elements whose *squares* are converging to 2 (that's how we constructed it), and the sequence itself keeps getting tighter and tighter (since it's a Cauchy sequence).
So, we gather up all other sequences of elements whose squares converge to 2, and put them in a set. That set is the real number "square root of 2".
Another sequence in the set is {2, 1.5, 1.42, 1.415, 1.4143, 1.41422, ...}.