Example, lets say you pick N = 3, then the statement refers to the terms: ( are just two of these terms, any two you want to pick)
If N = 126, then the statment refers to:
So basically the definition is saying: A sequence is Cauchy if after a certain point (marked by the N-th term), all the terms of the sequence are really close together (that is, within a distance of of each other), no matter which 2 you pick. It would basically mean that the terms are convergng to some point, and so, they will all get very close to each other as they all get very close to said point. You will learn (or should have learned) that a sequence converges if and only if it is Cauchy. That is a good context to think of it in.