My buddy gave me a set of LOOK clipless pedals and some cleats. I have to buy some shoes. I don't understand how they work. Are there any specific brands that I need to look for to be compatible?

You need shoes that have a three-hole bolt pattern in their bases. Any road-specific shoe should work. MTB shoes will not.

I would highly suggest that you try on shoes in local bike shops if at all possible. You can buy online, but you might be returning a few pairs of shoes that don't fit right. (Definitely do not settle for ill-fitting shoes. Your feet will hate you.)

Yeah, that's what I was thinking about going to the LBS to try on some shoes. I don't feel comfortable ordering online without seeing if they fit first.

Or you could give your LBS the business even if it costs 20% more than online.

And at this point I would suggest that you have a look at the Specialized Comp Road. It's a lot of shoe for $150. It is relatively narrow, though, so if you have wide feet it probably will not be a good choice. I looked at a lot of shoes and picked these, but I have fairly narrow feet.