How does one connect three monitors to one computer?

Take a look at Jeff Atwood's blog: Three Monitors For Every User; here he speaks of AMD's Eyefinity cards which can handle 3 concurrent display outputs. He then references Converting DisplayPort and/or HDMI to DVI-D? for that third output.

Eyefinity on Newegg

Two cards is not the only way, but for most people's needs, it is probably the best, or at least the most cost efficient (depending on whether you have an extra monitor already).
There is also the option of getting a USB monitor, but these will typically be smaller screens with lower resolutions and options, but depending on your needs and budget, may be better suited.
Last is the option of just replacing your video card with one that can handle more than two monitors on its own. This web page contains a list of cards, some of which can handle up to four monitors.