A cubie is generally any cube-shaped, solvable piece on an NxNxN puzzle. Each corner piece, edge or wing piece, and center piece is a cubie.
A corner piece is a part of the cube with three visible faces, each bordering three edges (or three corners for the 2x2x2). On every NxNxN cube (N>1), there are 8 corners. 
An edge piece is a part of the cube that has two visible faces, each bordering two centers, and either two corners, one corner and one edge, or two edges, depending on the size of the cube. Every NxNxN cube (N>1) has 12(N-2) edges. 
A center piece is a part of the cube that has one visible face. Each borders four edges, one edge and three centers, two edges and two centers, or four centers, depending on the size of the cube. Every NxNxN (N>1) cube has 6(N-2)^2 centers. 
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.