In my experiences with American student unions they're a bit like an airport, but without the messy business of having planes coming and going.
American unions tend not to have bars although a few sell alcohol in some areas (ours only in the bowling alley), usually this is only domestic 3.2 beer. Instead the two most frequented spots in the union are likely the bookstore and the food court.
Almost every single union I've been to has a campus bookstore selling more or less all the books and supplies needed by students along with various clothing, trinkets, and ephemera associated with the school. The sorts of things one might wear or take to a sporting event. Often they will also sell computers and small electronics as well or offer a seperate store for this purpose. This is typically in competition with an outside, off-campus bookstore, but it may not. At Kansas State University the bookstore was recently outsourced to the only other college bookstore and the only other place selling books of any sort in town (unless you count B. Dalton, Hastings, or the small used bookstore)in a rather monopolistic, but oddly unopposed move.
Likewise the food court tends to be a fixture providing a place to eat for off-campus students and staff who don't bring lunches or have an opportunity to leave campus. Depending on the school there is often a variety of school-owned and national chains providing about 6-7 choices in an average-sized state school. Taco Bell seems to crop up rather often likely due to being a student favorite (cheap and fast) as well as their strategy of putting the things in everywhere up to and including high schools.
Along with the above fixtures coffee shops, convenience stores, meeting spaces for campus organizations, conference rooms, auditoriums of various sizes, bowling, pool tables, arcades, banking (if only a line of ATMs), postal services, stationary services, etc. are all common. Many unions host bands, films, and other forms of entertainment, but it largely depends on who's in charge of the place and how large your school is. Even Erskine College, a rather small school in Due South, SC with only a few hundred students had a union consisting of mailboxes, a snack bar, and some pool and ping-pong tables.
Generally the student union is a place to stop off during the day between classes to run minor errands or purchase things related to the school. Occasionally people will hang out there or the school will utilize open areas for student functions (campus elections being popular for this), but it serves a good deal less as a student club and much more as a small indoor strip-mall owned by the school and catering primarily to off-campus students.