First, how much would you like to spend on the computer? Think of how you would use your computer, do you want to bring it with you and do you travel much?
If not, desktop is by far a better solution. Desktop computers are much cheaper. If you really want to save money, don't buy computers like Alienware. They are really overpriced. The best thing you could do is to build a computer yourself, but since you're just a newbie that might not be the best solution either. If you want to build your own computer here's a great tutorial, the tutorial also includes amazon links to all the parts you need: http://www.youtube.com/watch?v=BCjxMG4OLdE
When you're looking at good computers, the most important thing is the CPU and the Video card. Things like RAM and other stuff are important too.
For the CPU I would suggest a Intel core i5 or i7. Intel i7 is better but also more expensive so if you want a good PC you can even go with i5, they doesn't really matter in gaming anyways. When you look at the CPU's you should look at the gHz and the cores. The higher gHz and the more cores, the better.
Video cards are not as tricky. Good gaming video cards are for an example something from the nvidia GEFORCE GTX. The best available for desktop PCs is the gtx 690, but every video card that is gtx 560 and above can run all the latest games on high settings.
RAM (Random Access Memory) is for an example important if you run many applications at the same time, but if you only game you don't need more than 4 GB DDR3. If you're going to edit movies with a lot of complicated software like Vegas Pro 12, you can go for 8 GB DDR3 and it'll for sure run smoothly.
A computer also needs a cooler, motherboard and many more important stuff but when you're buying a built computer you don't have to worry about it.
I hope I clarified some things for you and that I understood your question right.