[Winga]

What's meant by chemically equivalent?

Does it mean that the protons have same electron environment or the protons can take part in same chemical reactions?

[movies]

In NMR terms it means that they are in an identical electronic environment.

[Winga]

Does same electronic environment mean the same e- density around the protons?

[movies]

In part, yes.  It also includes other things though.

Also you have to account for the NMR time scale, so the electron denisty might change but as long as it averages to the same thing over the amount of time that it takes for the NMR to detect a signal then it will appear to be equivalent.

[Winga]

Chemically equivalent should be related to chemical reactions, right?

e.g. propane, the methylene protons are chemically equivalent because no matter what substituent replaces either one of them, it will give a same product. This can also apply to methyl protons.

[movies]

Well, not necessarily.  If you were to look at a particular rotamer of propane the protons on the methyl groups would not be identical.  In fact, there is no single arrangement where all the methyl protons would be identical.  When you consider the arrangements over time, allowing for bond rotations, then things average out.

The "chemical" part of the NMR term is from the "chemical shift" which is determined by the electronic environment.  Say you had acetone and you wanted to deprotonate it to make the enolate.  In order to do that the proton you want to remove has the be parallel to the pi bond of the carbonyl group.  The other protons aren't acidic because the orbitals don't line up right.  In effect, they aren't equivalent in chemical reactivity, but they will coalesce to a single signal in the NMR.