Can a computer have 2 different graphics cards?

Yes, you can run run 2 different GPUs together in concert so long as you don't use an SLi/CrossFire bridge. One possible scenario is that you could use one for doing PhysX and the other doing all else.

You can run as many graphics cards as you have slots to put them in.

But... Why would you want to?

A graphics card is a multi purpose device. What you do with it is controlled by the driver.

The driver will operate differently , depending on the game or application.

You can usually attach two, or, sometimes three monitors to a single card if you want to.

If you will be gaming and simultaneously running a graphics design app, then you would want two monitors, but I don't think you could concentrate on two things well enough to do them at the same time.

I suggest you get a single good graphics card, and a single good monitor.
If your graphics design app needs a specific card from ATI or Nvidia, then that should dictate your choice. Games do well with either.

I wouldn't suggest it at all. Rather you sold of both the cards that you have and bought one card which is capable to doing both things perfectly.
Using 2 Monitors is not the best idea for gaming, it's usually 3 or an odd number. But it usually depends on the games you play, some of them might not even support multi monitor rigs, so, Instead of going for 2 GPUs and 2 Monitors, it's better to save on the price and buy a good GPU plus a bigger monitor if you manage to sell off 2 Cards and a Small monitor.
 

Two cards are only an option when you have one card already and do not want to sell it or let it waste but at the same time want to have little improvement in GPU, so you will end up buying the same card twice and will do a SLI or Crossfire.

In case you are making a new rig, then a high end single card is a much better option which will not only hv the high GPU but high GIGS as well.

remember ..

1 GB Card + 1 GB (second) Card == 1 GB Card with better GPU ( and not 2 GB)

so,, better go for a straight 2 GB single card.

really i'd say depends on his application... if he's doing alot of CAD or other professional software usage then yews it s agood plan as he might need a pro series card which cost alot mroe than gaming cards usually, liek a firepro or a quadro

then for gaming you can get a card better suited for gaming a gtx or a radeon hd

people on here so far are talkgin about gaming only card... they aren't realising that a proffesional workstation card and a gaming card are COMPLETELY different

As the OP stated, He's Just Beginning to Study...... Graphics Design....  
And as he is One bored gamer........ it's pretty obvious.....

But he has better options for lesser cost if he goes for the High end GPUs.....
Here's a direct quote.......
"The best option, Soft moding Geforces and Radeons to Quadros and FireGLs. You get the gaming performance when you want it, and the workstation performance when you want it, all for gaming/consumer prices.

It's pretty easy for Geforces (FireGLs are trickier, and IMO only worth it if you know the FireGL is significantly faster than the Quadro equaivalent);
http://www.techarp.com/showarticle.aspx?artno=539

Just don't use this for jobs that require certified workstations cards from ISV, IHV or contract. But for pro-sumers dabbling at home and such you get near 9x% of a Quadro for usually 25% (or less) of the price. "

And here's the whole topic...
http://www.tomshardware.com/forum/243795-33-quadro-gefo...

It's not that I disagree with you goofy, it's just a little too early for him to switch to Quadro's and Professional Cards , this is something I feel, maybe wrong, but starting and experimenting at not a very high price is a better option I believe....

i was basing his statment of

"Can I run 2 cards in one machine? One dedicated for design and one for gaming"

i firured it'd be a workstation card and a graphics card fo rthe same computer... but yes if he's not going to run a workstation card and instead its two gaming ones totally just get one fast gaming card and use software workarounds for optimisation

the article there that you linked is quite good

http://www.techarp.com/showarticle.aspx?artno=539