The Tenth Amendment ensures the states have the right to pass their own laws, which have not been delegated to the US Government.
http://en.wikipedia.org/wiki/Tenth_Amendment_to_the_United_States_Constitution
Unless the power, jurisdiction, or right has been delegated to the United States Government, it is left to the states.
http://wiki.answers.com/Q/What_is_the_meaning_of_the_10th_Amendment
Back up to your OP for a minute and I think we can resolve this confusion. The 10th Amendment has no bearing on this particular ruling. You quoted it so you should know that. It says powers not enumerated to the federal government are given to the states (or the people.)
Immigration IS a power granted to the federal government (read Section 8 "uniform rules of naturalization"). Accordingly, how is the 10th Amendment even relevant here?
Let's have a quick primer on federal versus state power in the Constitution.
There are powers enumerated to the Federal Government. Some of those powers are specified as expressly prohibited to the states, i.e. printing money, declaring war.
All powers not explicitly mentioned are powers exclusive to the states, per the 10th amendment.
So we have 3 categories of powers.
1. Powers exclusive to the federal government because they are granted to the federal government and are expressly prohibited to the states. Only the federal government can make laws in these areas.
2. Powers shared by both because they are granted to the federal government but NOT prohibited to the states. Immigration is one of these by the way. Either the federal or state governments, or both, can make laws in these areas.
3. Powers exclusive to the states. Per the 10th amendment, this consists of all powers not enumerated to the federal government. Only the states can make laws in these areas.
For category 2 situations, the Constitution has something called the Supremacy Clause, which says if both the federal and state government choose to enact laws in one area, the federal laws trump the state laws to the extent that they conflict.
The case you link in the OP is a category 2 situation. Immigration can be regulated by either the federal government or the states. The court upheld the Alabama law because it didn't conflict with anything in an existing federal statute so there was no problem with the supremacy clause.
The 10th Amendment is irrelevant in this case because that is what establishes category 3 powers - those that are exclusive to the states. Your case, however, is a category 2 situation and hence it has nothing to do with the 10th amendment.
- wolf