In my view, the "gain without closing the loop" point is somewhat hair-splitting. I think, the loop gain is still there, when you close the loop, but you don't see it directly. The expressions given by barry (except for a typo with closed loop gain) are however very clear about the problem, and it's possibly better to think about their substance rather than about words.
Open loop gain A is clearly the gain of the amplifier itself, without external circuitry. It's a number written in the datasheet.
Loop gain Aβ, the gain around the loop, has been visually described by barry, don't know what I should add.
Closed loop gain finally is the signal gain of the feedback amplifier, Vout/Vin.