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.

you can also do this with triplehead2go available at newegg fairly cheaply if you have a supported video card already(the card doesn't necessarily need to be on the supported list - in my experience matching resolutions and refresh rates was enough).

With DisplayLink hardware you get DVI over USB which you can then attach a monitor to. Their software allows up to 6 monitors attached in this way.

I have one and liked it for my third monitor. For programming it is great (as you do not have intense screen updates like with movies etc).

http://www.displaylink.com/shop/index.php?product=5

Just buy a graphics card from ATI labeled "Eyefinity Edition". These cards range in price (and quality) from 50$ to 500$ so you can pick a model that suits your needs. They have multiple (up to 6) mini display ports and you can plug in a monitor in every one of them.

The software allows you to run several configurations like viewing different content on every monitor or stretching a image/video/game across all monitors.