Well, for power, you'll definitely want a desktop. One of the biggest things is the video card, so you'll need to make that a priority. Next you'll want to focus on the CPU. It should at least be a dual-core, and a quad-core is nice if it won't break the bank. Third, you'll want to look at RAM. 4GB+ of DDR3 RAM should be fine. Finally, you'll want a power supply that can handle everything.
Now for some details.
As far as video cards go, the numbers can be a bit confusing. For example, 820 is a higher number than 770, so which one is better? It's actually the 770. The easiest way to figure this all out is to just Google the video card and find some benchmarks for popular games so you can see how they perform. AMD and Nvidia are both good options. AMD tends to be cheaper, while Nvidia has a better reputation. I'm not really too knowledgeable about them. I'd say get a card with at least 1GB video RAM that's at least DDR3. GDDR5 is super fast, but it's expensive.
Next, the CPU. As I said, dual-core minimum, quad-core preferred. As for clock speed, anything above 2.5GHz should be fine, really. 3GHz+ is ideal, however. Again, AMD is cheaper, but Intel CPUs have some nice features like hyperthreading. Go with what your budget permits.
RAM is pretty easy. As I said, go for 4GB+. With, say, 8GB I guarantee it'll perform well for a while, and anything over 8 is overkill, to be honest. Definitely go for DDR3 RAM over DDR or DDR2. Try to find some that's at least 1066MHz.
Finally, the power supply. A powerful video card will draw a lot of power, with the CPU also being a fairly big power consumer. Check out
this page to get an estimate on what kind of wattage you'll need.
Hope that helps