Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...
Lots of definitions of "respect" available, but I'm pretty sure this is the one they were shooting for:
re·spect
To feel or show deferential regard for; esteem.
So, the point is to have Congress avoid making laws that are pro-one religion or anti-some other religion.
Is that a "seperation of church and state"? Not entirely.
For example, the Supreme Court could hand down a decision based solely on their religious views. It would not violate this clause - since the clause refers to Congress.
Additionally, the President could veto a bill on solely religious views and likewise not violate this clause.
However, there's clearly more going on here. What's being said is this "The Government should not be taking sides in religious debate."
Likewise, the Government should not be handing money to one group or people based on their religion, they shouldn't be giving people a pass to break the law based on their religion, and they certainly shouldn't be twisting the laws of the land to fit into a specific idiology of an individual religion.
And, while the true power of the US Government SHOULD reside in the Congress, it has steadily been pulled toward the Executive branch.
So, when people complain about seperation of Church and State, what they are really saying now adays is that they don't like how the unconstitutional powers of the Executive are being steered by non-elected officials who cowtow to extreme religious groups.
After all, Ashcroft ANNOINTED HIMSELF IN OIL upon taking office as AG! That's more than just a little weird.