Nonstandard analysis is the name given to the weird cross of analysis with mathematical logic and model theory which lets you use infinitesimal quantities correctly.
Nonstandard analysis (NSA, for short) is the name usually given to one of several reformulations of Leibniz' infinitesimal formulation of calculus. Leibniz' (and Newton's) concepts for infinitesimal calculus are deeply unsatisfying: they refer to illdefined abstract entities; attempting to manipulate these infinitesimals can yield correct proofs when done in the "approved manner", but very similar manipulations will let you prove absolute nonsense. And since there is no formalisation of these concepts, it is impossible to distinguish right from wrong.
In the 19th century, the work of many mathematicians, including Cauchy, Bolzano, Dedekind, Weierstrass and Cantor (in no particular order) led to the first formalisation of calculus  the δε ("deltaepsilon") formulation hated by 1st year students ever since. This has enjoyed spectacular success: it lets you express all the concepts used in calculus, yet all its concepts are welldefined (indeed, it helped clarify concepts such as continuity, splitting them into subcategories such as uniform continuity, absolute continuity, and the like; even Cauchy confused uniform with plain continuity before this was formalised).
Along the way, this "standard" formulation of calculus lost the concept of an infinitesimal. Indeed, the very concept is selfcontradictory (see smallest number greater than 0 for some examples of the confusion that ensues). On the other hand, the infinitesimal formulation of calculus has certain appeal. For instance, it generally requires one quantifier less than the "correct" formulation.
During the 1960s, logician and applied mathematician Abraham Robinson used model theory in a somewhat peculiar manner to obtain models of calculus which contain objects worthy of the name "infinitesimals". Other mathematicians (notably E. Nelson and Luxemburg) proposed other techniques (also grounded in model theory, set theory and mathematical logic) yielding the same or similar objects.
These techniques share a general structure. The standard model of real numbers is extended to a larger "nonstandard" structure. This larger structure is proved to have several useful properties. The most important property is that the extension is conservative: if you can prove something in the larger structure, it is in fact true in the smaller structure^{footnote 1}. Thus the general plan is to use peculiar properties of the nonstandard model to prove something is true, then conclude that it is true in the standard model, too. The nonstandard model can be viewed as a crutch to prove the theorem; it is an invisible mechanism which does not appear in the formulation of the theorem, only in its proof.
See NSA: Introduction and Construction for the start of my extended notes on NSA.
About these notes
These nodes are my attempt of a reasonably nontechnical description of how NSA works. Much more can be said about the mathematical logic background of the subject. These notes follow Moshé Machover's presentation (mostly repeated in Bell and Machover's A Course in Mathematical Logic, chapter 11). Machover's technique is to use the tools developped by Robinson to extend ZF set theory itself; this means all higher level objects (such as sets, functions and Banach spaces) already appear in the basic theory, and no further forays into the dark recesses of model theory are required to manufacture them.
You should have seen the basics of the δε formulation of calculus, to have some idea of the concepts involved. It doesn't matter if your attitude towards δε is one of love or loathing, but the more exposure you've had to analysis, the better. Background in mathematical logic and set theory won't do you any harm. In particular, friends and relations of the compactness theorem and ultrafilters pop up all the time; I shan't always be pointing them out, but it's always nice to see an old friend...
The source for the nodes are my notes from a course on the subject taught by Moshé Machover in the Winter 95/96 semester at the Mathematics department of the Hebrew University in Jerusalem. I have omitted as many technical details as possible, in the hopes of making the subject somewhat more accessible. Any errors are most likely due either to these simplifications or flaws in my notes; they are of course solely attributable to myself.
For a more complete treatment, see Bell and Machover's book. Be warned, however, that that book is very technical!
About other nonstandard analyses
A completely separate way exists to generate "infinitesimals" in a rigourous way. This one involves the use of asymptotes (of real functions). By comparing rate of growth of various (usually positive monotone) functions, we can get a nice field which doesn't satisfy the archimedean axiom. A bit more work lets us pass some "small" functions off as infinitesimals of various orders.
But I find this style very unsatisfying. A good deal of analysis is required to make it work. For a supposed replacement for analysis, this doesn't bode well.
On the other hand, not having to lug about a huge amount of mathematical logic can be very convenient.
Outline of the rest of the notes
 Construction

 Terms

 Sample applications

 Helpful sidenotes

All are referenced by the main text above: