A cubie is generally any (seemingly) cube-shaped, solvable piece on an NxNxN puzzle. Each corner piece, edge or wing piece, and center piece is a cubie.
Corner
A corner piece is a piece that is on at least three faces. All NxNxN cubes, when N is strictly greater than one, have exactly eight corners; all dodecahedrons have 30 Corners.
Edge
An edge piece of a puzzle is normally a piece that has two stickers (because it is part of two faces). On the 3x3x3 there are 12 edges, and for each pair of adjacent colors there is only one edge with those two colors. On larger cubes (such as the 5x5x5), however, there are several edge pieces for each pair of colors.
Centers
A center piece of a puzzle is a piece (also called cubie for a cubic puzzle) that only has one sticker and lies on only one face. On the 3x3x3 there are 6 centers, one for each color, which do not move. On larger cubes (such as the 4x4x4), however, there are many center pieces of many different types. Every NxNxN (N>1) cube has 6(N-2)^2 centers.
Cores
Cores are the parts of the cube with no visible faces. They control the movement of the cube. Each NxNxN cube (N>1) has a core which takes the space of (N-2)^3 cubies, though usually they are blended together into one mechanism, which sometimes also blends into the unseen faces of the corner/edge/center cubies. For example, the 2x2x2, which should have a core of 0 cubies, has to have one so that it can function. This means the 2x2 core is hidden inside the puzzle.
