Series vs. Parallel: A Primer
Many of the mods discussed throughout the site create differences in tone by altering the way two or more (but usually just two) pickups are combined together in a guitar.  Although there are other issues involved such as the phase of the signal, basically there are only two ways pickups can be combined: in series and in parallel.

Parallel connections

 Basically two pickups are independent of one another, but then send their outputs to the same place.  Almost all guitars combine their pickups in parallel.  Although I drew individual single coils, these could just as easily be a pair of humbucker (which would likely have series connections within them; see below).

Series connections

 The most common series connection is between the two coils in a typical humbucker.  One coil (or pickup, depending on the guitar) leads directly into the other in a train before reaching the output.

Understanding the math
If you've already had a decent high school physics class, this is nothing more than the rules for adding resistors in series vs. in parallel.

Let's say you have a couple pickups with an impedance of 5kOhms each.  If you select an individual pickup, it's just 5k.

Series: Now if you combine the two pickups in series, the math is easy.  Series additions are simple addition.  In this case, 5 + 5 = 10.  The output is double each pickup on its own.

Parallel: If you combine the two pickups in parallel, then the math is a little bit harder to explain in words.  Basically it's the sum of the reciprocals.  So the reciprocal of 5 = 1/5.  Add them together (1/5 + 1/5= 2/5).  Take the reciprocal of that: 2/5 --> 5/2 = 2.5.  The output is halved what each pickup was by itself.

How's it sound?
It depends on what you start with, but in general, a parallel combination is somewhat weaker feeling owing to it's weaker output (i.e., less impedance).  This is why (again, at least in part) if you're playing a solo you switch to one pickup or the other instead of both.  Similarly, if you want a thrash metal sound, odds are you're going with the bridge pickup.

By contrast, a series combination thickens the sound up, hence humbuckers on average have a higher output than most single coils.  Output isn't the whole story though.  Any series connection means fewer high frequencies make it through all the extra coil windings, so things sound "darker" (i.e., not so "bright").

Note: These are generalizations.  For example, if you add two single coil pickups together in series, you aren't going to get a very good humbucker, just a thicker sound than you started out with since most single coils are lower than the impedance of either of the individual coils in most humbuckers.  Conversely, a coil tap gives you a single coil, but it doesn't have exactly the sound people expect from a single-coil pickup (e.g., a Strat or Tele) because that coil almost certainly was made with a higher impedance already.