The principle of superposition is that any addition of quantum states is also a valid quantum state. For example, you can have a system that has a state which is both "the particle is at x = 1m" and "the particle is at x = 2m". What really happens physically when you measure the position of the particle is not specified by the theory, but the Copenhagen interpretation says that one of those positions will be chosen depending on the coefficient in front of each state, e.g. if we have a state |psi> = 1/sqrt(2) |x = 1m> + 1/sqrt(2) |x = 2m> then we get either 1m or 2m with 50/50 probability. Then if we measure 1m we will find that afterwards the state will be |psi> = |x = 1m>. Again, what actually happens during this process is not well understood by physics yet, and also is one of the things that differentiates different interpretations of quantum mechanics, so yes that is "philosophy".

> The cat is dead and alive because you just don't know it. But it surely is one or the other, isn't it?

The Copenhagen interpretation says it is both dead or alive, or that the above particle really was in two places at once. Other interpretations, such as the Many World Interpretation (MWI), say different things. For example, MWI says that upon measurement the universe actually branches into many universes, one for each possible outcome, and that we live in one such universe (the universe with a living cat, for example), but that the state itself is not ontologically a mixture of those two orthogonal possibilities.

Entanglement is a very different beast. Basically it means that two systems interact such that measurements on them are highly correlated, i.e. by measuring something on system A you immediately know something about system B, even if they are very far apart. Again, how you interpret entanglement (i.e. whether or not "spooky action at a distance" actually happens) depends on your interpretation of quantum mechanics.