Cafes generally refer to small, casual eateries that are a step above fast food in quality and service, but not quite as formal as a traditional sit down restaurant. Some cafes offer table service, some offer more of a take out style of dining.
There are literally thousands of places to drink coffee in the city, from the corner deli to elegant European style cafes with violin players. A "coffee house," in the old-fashioned sense, is a place to sit, relax, sip a good cup of coffee (or tea) with a pastry, read the paper or chat with friends. Sometimes, people even study, work or play games such as chess.
You can nearly always find a pretty decent cup of coffee with or without caffiene, hot or iced.