My personal setup.
Nvidia supports triple monitors with the name "Surround", opposed to AMD's "Eyefinity". I've used both with an AMD 7950 and Nvidia 670 graphics card. Let's start with the obvious differences.
AMD:
Newer AMD cards usually have ports like the image on the bottom.
Of course there are other port setups, but this is the most common with 7950's and some others in the 7000 series.
Pros:
*You get to use an AMD card, which is usually cheaper than Nvidia for equal or only slightly less quality of comparable cards.
*Variety of ports for flexibility.
*AMD's great video control panel interface
Cons:
*Mixing the displayport with the DVI or HDMI (Which you HAVE to do in a triple setup) you WILL experience screen tearing, its literally impossible to avoid it because the differences in signals.
*Mini-displayport adapters are expensiveeeeeee!
*Eyefinity's warping of Win7
My Opinion:
The newer AMD cards that have Eyefinity are excellent products but I'd recommend you find a model that has only a combination of DVI and HDMI so you avoid screen tearing. Also, using Eyefinity on Windows 7, you're toolbar is stretched across the three screens, which I don't really like, but other people do. Overall, the newer AMD cards are a good buy for triple monitors, and you only 'need' one! Unlike previous generations that required cross-fire, these newer cards can run all 3 monitors on 1 card, like a champ.
Nvidia:
Newer Nvidia cards will usually have ports like this.
Again, there are other port setups, but this is a common setup at the moment.
Pros:
*Displayport isn't necessary, no screen tearing!
*"Surround" is more aesthetically pleasing, in my opinion
Cons:
*Never liked Nvidia's video control panel
*More expensive cards for a questionable amount of performance improvement against their AMD counterpart.
My Opinion:
Nvidia is the Coca-Cola of the GPU world and AMD is the Shasta. They are excellent cards and Surround works great! Is it THAT much better than an AMD counterpart? In some ways yes and other ways no, but overall their quality is superior, if only slightly. And I just love their "Surround", it makes the windows toolbar stay in only the main screen, making everything look sleeker.
Conclusion:
This review probably seems a bit bias towards Nvidia, but I really have had both cards for extended periods of time and love them both. They can both run triple monitors, full settings on most games, like a beast. It'll really come down to your budget and it's a big decision so don't take it lightly!