Consider the sets [a,b) for any a and b in the reals (and also allow b to be infinity).
A borel set is then something that may be obtained by repeatedly using the operations if union, intersection and complement to these sets and any sets that we obtain in the process too.
ok, sounds hand wavy and uninformative. sorry. this is called expressing it in terms of a "basis".
It is easier for me to put it this way: essentially every subset of the real numbers that you an describe is a Borel set. I hope I don't saty something false here, but the only way you can define a subset of the reals that is not a borel set is by using the axiom of choice, and we can perhaps think of this as being "pathologically" bad and not a representation of any set you'll meet in "real life".
here is a link showing just how hard it is to define a nonmeasurable set.