Click here to read the entire opinion. But let me explain it, in a nutshell.
There were four provisions at issue in S.B. 1070:
1. Arizona made it a misdemeanor for being in the United States illegally.
2. Arizona made it a misdemeanor for an illegal alien to seek or obtain work.
3. Arizona allowed its police to arrest a person without a warrant if the police thought they had done something that could have them "removed" by immigration officials from the United States, i.e., being an illegal alien.
4. Arizona allowed its police to verify immigration status during a routine stop or investigation.
The US Constitution has a Supremacy Clause by which federal law "preempts" or "trumps" state law. So, the Court held today that provisions 1 through 3 are pre-empted by federal law since such measures exist in federal law. Mind you, this does not address the issue of what to do when the federal government does not enforce its own laws, but from a purely legal aspect, if the federal government has laws that govern something under its responsibilities (immigration), no state can override it with one of its own. So they are struck down.
But the Court upheld the fourth provision. That provision - also known as Section 2(B) of Arizona law S.B. 1070, requires that state officers (police, sheriffs, troopers) make a "reasonable attempt" to determine the immigration status of any person during a routine stop, investigation, or detention, if "reasonable suspicion" exists that the person is an illegal alien.
So - carry your ID and be prepared to answer questions truthfully. Oh, and obey the laws so you lessen your likelihood of being stopped. Good advice for natural-born citizens, naturalized citizens, legal residents, and visiting foreigners.