The law's purpose is based off of a federal law that prohibits illegal immigration. Contrary to popular (actually, minority, because the majority of the country supports it) belief, its parameters strictly and directly prohibit racial profiling of any kind. Police officers aren't allowed to go up to anyone they want and say "show me your papers." In fact, they are required to check immigration status if one has already been arrested for a crime.
Countless other states have introduced and/or instituted their own versions of SB-1070. The Obama Administration has shown blatant opposition to the bill and managed to essentially take away all proponents of it that gave it any effectiveness at all. Not only that, but State Department officials admitted to dissing Arizona's "humanitarian rights-violating" bill to countries such as China and Libya. Most, if not all of the Administration's accusations of the bill have truly been completely false.
Someone who has actually read the Arizona Immigration Law: "No, you couldn't. Unless going out for ice cream is a crime now."