What is the process of legal immigration into the U.S.?

Arizona recently passed a law basically saying that if you commit a crime (say even a speeding ticket) and they have reason to believe you may be an illegal immigrant they can ask for your proof of residency. Apparently there is a huge fuss over it, and many people think it is horrible.

I’m kind of confused because isn’t this just making it easier to enforce a law that is already in place? Why don’t people just immigrate legally? What is the process to do so? Is it difficult? Expensive?