The orbit of a piece is the set of all positions where it can be sent to by normal turns of the puzzle (not allowing cube rotations). This is distinct from the set of positions with similar-looking pieces - in some puzzles, it is not immediately obvious where a piece can be moved to.
The cube term is an application of the definition of "orbit" in group theory.
There are also some cases where pieces look similar but do not actually belong to the same orbit. One example is the oblique pieces on a 6x6x6. There are 48 obliques, but they are actually in two separate orbits of 24 pieces each, and pieces from one group can't move to the other position.
In a standard twisty puzzle (such as a 3x3x3 or a Pyraminx), the whole puzzle obeys a specific kind of symmetry called Polyhedral symmetry (see Wikipedia) which limits the number of positions that can be in a single orbit. For tetrahedral puzzles like the Pyraminx, an orbit can have at most 12 positions; for cubical or octahedral puzzles like the 3x3x3, an orbit can have at most 24 positions; and for dodecahedral or icosahedral puzzles like the Megaminx, an orbit can have at most 60 positions.
Use in BLD Solving
Blindfold solving is in most cases based around solving pieces in cycles - moving a small number of pieces to each others' places without affecting the rest of the puzzle. When the puzzle is solved, each piece has to end up somewhere else in the same orbit. This means that when memorizing the position of a puzzle you can consider each orbit separately, and determine the cycles of pieces within one orbit before even looking at the next one. This fact, combined with the maximum size of an orbit, means that every piece on even the largest NxNxN cube can only be in one of 24 places, so we can memorize each piece by a single letter for any size of cube. Thus the same kinds of memory techniques can be used on any size of cube, and the biggest difference between different sizes is the amount of information.