If we say the universe is expanding, shouldn't it be expanding relative to something? I don't understand, if everything in this world is relative to something else, then cannot we essentially say that nothing exists independently? We say that the universe is considered to be the ultimate 'background'. However, if we say the universe is expanding, shouldn't it be expanding relative to something? 
 A: The universe is expanding, in the sense that things in it are getting farther apart. It is not expanding into anything because it already is everything. There simply is nowhere else to expand into.
Lets knock it down one dimension. Your universe is the surface of a balloon. The balloon is slowly being inflated. Your universe is getting bigger but nothing else is getting smaller (remember, you are unable to leave, look from, or perceive anything that is not on the surface). The only thing you can measure is points are further apart than they used to be.
Classical mechanics don't really work at the two extremes: the quantum level and the whole-universe level. If we ever fully understand the whole process I expect we will find that the complete equation applies across the board, but certain factors are negligible at human-perception levels. Motion is a good example here: we don't need relativity to calculate driving times, even though my watch does slow down when I drive to work.
A: This is the specific definition of expansion (which personally dont agree). That the universe is not expanding wrt to sth (e.g time or space), but it is in fact time and/or space (or more correctly space-time) which expands (according to the theory of Big Bang, et al.).
A: Expansion can be orthogonal to real space.  For example, if you blow up a balloon, the surface expands relative to a point not on the surface.  What could be happening here is that the ratio of length to arc of space could be getting bigger, so that a foot occupies less arc than before.  
Eventually, we will end up eg 3 ft high, but the mile remains a mile.  Space would have expanded because we would judge ourself to be 6 ft, and the mile would then appear to be two miles.
