1) No, the tablet works directly with the computer, and what you draw is dependent upon the program you're using and any number of variables. At a basic level, it becomes another mouse, but one that has a 1:1 relationship with the tablet area and your screen (put the stylus on the top left, the cursor jumps to the top left of your screen). Obviously there are other perks other than it just controlling the cursor, such as pressure sensitivity and so on.
There is the option of a tablet PC for portability, which is a laptop with a tablet built into the screen, but those are fairly expensive.
2) Anything and everything can run a graphics tablet, though the suggested requirements should be printed on the box and on the products' websites. Generally, any computer with Windows XP or later, or Mac OS X will be able to run a tablet without trouble.
3) Wacom, mostly. I've heard good things about Genius tablets, and they're very inexpensive, but the Wacom line is generally a lot more reliable and generally a good bit better. You may want to look into the Bamboo Fun, or if you have a bit of a higher budget, an Intuos4. The Intuos series has twice the pressure sensitivity along with tilt sensing for things like spray brushes, which makes them a little more versatile; Plus, the pens seem to last much longer (not to mention they have replaceable nibs). Personally, I have an Intuos4 Small, and with it being my first tablet, I think it's probably going to last me a good while to come. It's also surprisingly roomy for a "small", but one thing you should consider is that the larger versions will have somewhat more precision, as there's more area to translate to the same screen space.
