SR4A is absolutely essential, just as the PHB is essential in D&D. Every player should at least have a digital copy, though I personally have digital and hard copies.
Arsenal, Augmentation, Unwired, Street Magic, and Runner's Companion are awesome books that expand upon several basic parts of the world, and the rules. You should have at least one copy of each book at your table.
Setting books. Runner Havens and Seattle 2072 if you're in Seattle, for instance. In addition to telling you the layout, and showing who's who, it provides a nice list of hangouts divided by district.
Attitude is a must, from a fluff standpoint. The whole book just breathes life into the Sixth World, and some of the parts are extremely useful for teaching new groups, or for refreshing the memory of old hands. The gear is nifty. If you can't figure out a use for color changing clothes, you're not trying hard enough.
I'd have a copy of Sixth World Almanac at your table, too, to pass around to people now and then, so they get a picture of what the world is like.
Everything else you can leave off, and not miss too much, but I love them all.