Let's talk about the topological definition given by Evandar a bit more. Topological spaces are rather abstract, but the ones that we're mostly interested in are metrisable, which means they can be made into a metric space. You should think of a metric space M as a set of points together with some sort of way of measuring the "distance" between any 2 of them. This "distance" is just a function from MxM to the positve real numbers that satisfies certain simple axioms we want the distance to satisfy. The real number line, with the distance between 2 points x and y given by |x-y|, is a typical example.
Now "countable" means "small" in mathematics, for various reasons. The real numbers are not countable, but some subsets of them are, for example the integers or rationals. It's the rationals we want to think about here, because they are also what's called dense. For real numbers, this just means that in any interval you can always find one. In a general metric space, a set is dense if every point in the whole space can be written as the limit of a sequence of points, each element lying in the dense set in question. So every real is the limit of a sequence of rationals (just truncate the decimal expansion). It's not too hard to see this is equivalent to the first thing I said.
Dense means "large" in some sense, because every point in the whole space is as close as you like to a point in the dense set. So if a set is countable AND dense, it's both "large" and "small". The interplay between different notions of the "size" of a set is fundamental to lots of analysis.
Not all metric spaces are separable, for example the set of real numbers with the "discrete metric". So if a space does happen to be separable, it's really useful, because we only need understand a "small" (countable) set to understand lots about the whole space. This allows us to prove theorems more easily. For example, these ideas lead to a proof that if f is a continuous function on a closed bounded interval and if the integral of (f(x) multiplied by xn) over this interval is 0 for every n, then f must be identically zero. This is known as the moments problem.