I use beer a LOT for my cooking. As the others have said, you need to watch out for the bitterness, which can intensify when you cook and reduce it down. As such, I often cook with different beers than I drink.
You can use the IBU (International Bitterness Unit) rating for a beer to get an idea of whether your favorite beer will be a problem.
Personally, I use Guinness and other dark beers when red wine is called for in a recipe. The flavors are definitely different than the red wine version, but it usually comes out good. I also cook quite a bit with wheat beers and the kind of beers that people describe as "nutty", like porters.
America's Test Kitchen looked at beers for cooking at one point and a "non-alcoholic" beer actually did really well: O'Doul's Amber. I agree with them and tend to keep it on hand for cooking, even though I never drink it. It stays mellow and away from the bitter flavors when used for cooking.
So, what do I cook with it? Like most of the others, cooking bratwurst before grilling and cheese fondue, but I also use it for lots of other stuff.
I have a baked bean version that uses 4 cans of beer and a couple of shots of whiskey and sits in a crock pot on low for 24 hours. I've made this with all different kinds of beer and each brings something different to the results. (Recipe for J's Drunken Beans)
I also like using beer in most of my stews. Throw beef chunks, veggies and beer in the crock pot in the morning and it's a tasty stew when you get home.
It works well as the liquid for braising pork or beef. In the winter, I'll often throw a pork shoulder roast into the crock pot with a bottle of beer and let it slow cook all day. When it's completely falling apart, I pull it out, shred it and add back as much of the beer as the shredded mess will hold (which turns out to be far more than you thought). What results is a juicy sandwich option, which is really good, topped with mustard or bbq sauce. It comforts me when it's too cold to deal with slow smoking pork (I live in MN).
I'll also often take some of that shredded "drunk" pork to my baked beans and give the beans some extra dimension.
We do a version of "bangers and mash" that I use beer in the gravy/sauce that I like.
Beer and self-rising flour make for a really quick quickbread that has an interesting flavor profile. I particularly like using wheat beers like Leinenkugel's Sunset Wheat, as the wheat flavors work well in a bread.
Beers (and hard ciders, which I actually REALLY like) work well for marinades.
Beyond that, I tend to find myself just tossing a few ounces of it into a lot of other liquids/sauces to add some flavor.