It's probably best to start by having a look at wave-particle duality, and from there the Schrodinger equation, and then quantum superposition.

The Schrodinger wave equation fully describes a quantum state. If for example we want to extract position data for the quantum 'particle', we are presented not with a fixed point-like position, but rather with a probability distribution, a sort of smearing, in effect. This does not mean that there is say 60% chance that the particle is at point A, 40% chance it is at point B... because we are talking about a wave function here. The 'particle' is in both places at once, it's not that it's in one but we just don't know which, it is effectively in both but with differing levels of concentration. It is only upon taking a measurement, interacting with the system, that the wave resolves to a point-like particle, and it slots 100% into one of A or B. I'm obviously simplifying in this A-or-B case, but this is superposition; those two classical outcomes are both present in the quantum state.

Classical computers use bits, each bit can be 0 or 1. Quantum computers use qubits, where each qubit is in a superposed state of both outcomes.
So the classical bit is either 0 or 1, whereas the qubit is both, simultaneously. We can then see that whilst classical processing power scales linearly with each new bit, the power in the equivalent quantum system scales exponentially, 2ⁿ.

You might object that the quantum system still resolves into the same number of classical outcomes. And that is true. However here we encounter another key basis of quantum computers: entanglement. Those quantum states can be tied together.

In a 3 bit system, with 8 outcomes {000,001,010,011,100,101,110,111}, the classical computer can only be in one end state. So can the quantum computer. The difference is that the classical computer took one path to get there. The quantum computer, if the qubits are entangled, can take all 8 at once. This is why QCs are so great at problems like prime factorisation, which brings us back to Shor again.

I would say that observation isn't necessarily physically observing, it's any interaction with the environment that can trigger wave function collapse. We also get into a discussion of whether the quantum wave function is a 'real' thing, or merely a mathematical model to describe the underlying reality. Quantum mechanics has some solid maths behind it; the big problem is making sense of what that maths means using our poor human brains, which aren't tremendously well suited to the task. I don't think anyone fully understands QM.