Non-standard analysis is the usage of Model theory to study analysis. Since studying the saturated model of a theory is easier then studying other models, non-standard analysis studies the saturated model of theories with many symbols thrown in to make sure results are applicable.
One kind of elements that are in the saturated model are infinitesimals. Since it is consistent for a real number to be smaller then any finite subset of {1/n| n natural}, there is a non-standard real number smaller then all of them. In fact, there is a whole ideal of non-standard real numbers. If we start from the rationals, rather then the real numbers, and divide the ring of non-standard finite rational numbers by the ideal of the infinitesimal rational numbers, we get a field (because it is a maximal ideal) -- the field of real numbers. This kind of easy ways to get results which are hard work in classic, epsilon-delta analysis is typical. For example, proving that the composition of continuous functions is continuous is much easier.
There are not many results proven first with non-standard analysis. One of them is the fact that every polynomially compact linear operator on a Hilbert space has an invariant subspace, proven 5 years before classic functional analysis techniques were developed that deal with such problems.