Thumb Under: Normal way to play octave, like connecting in all the playing, 1245(then thumb under the thumb) 1245.
Thumb over: it's not actually over, just as a contrast to the thumb under method.
it's like play the 124, than land the whole hand on the first note of the next register again and play 124
for eg
C maj octave
124(CEG)124(higher CEG),124124124124....
This method is widely used, much more than you think
for eg.
in Ocean Etude(Op.25 No.12), if you don't play thumb over, you are actually torturing yourself(and the composer intended to play that in thumb over)
also the Presto Con Fuoco part of the Ballade 1, the 1,25,1,25 part, you have to play thumb over in order to play fast and accurate
For the last comment saying thumb over is not accurate.
Accurate or not depends on how much you practice and where you use them
FOr eg. if you play ballade presto con fuoco with thumb under, you are kidding yourself.(I played it.)