What actually is the Born-Oppenheimer approximation?

It can actually be different things to different people.
The same applies to the "adiabatic" approximation.
There is a nice old article from 1977, What does the term "vibronic coupling" mean?.
It clearly distinguishes different approximations that often get called "Born-Oppenheimer" such as crude adiabatic, Condon, Born-Huang,...
Then there are "corrections" to "Born-Oppenheimer" such as Herzberg-Teller.
The paper is useful because it clearly defines everything and has two nice Tables. One gives the relevant equations for the different approximations. Another compares the actual terminology used by different authors (pre-1975).

Does it matter?
Besides Born-Oppenheimer breaking down near conical intersections it also matters for

1. "intensity borrowing" where "forbidden" electronic transitions gain oscillator strength by coupling to the vibrational degrees 
2. electronic isotope effects: isotope exchange reactions can occur [at the level of the BO approx. the isotopes have the same potential energy surface].
3.  Resonant Raman scattering.

I thank Jeff Reimers for bringing the article to my attention.