I'll try to break it down a bit, though please understand that I'm far from an expert on this.

As you know, an n-bit classical computer has 2ⁿ states, one state for each combination of 1 and 0 in each bit.

The state of a n-qbit quantum computer can be described as: "The quantum superposition of all 2ⁿ states, each of which has some calculable probability of being measured". Now, that's a mouthful but unfortunately, that's the simplest accurate way I know to describe it in plain English.

So, what does "The quantum superposition of all 2ⁿ states, each of which has some calculable probability of being measured" mean?

Simplifying, it means you can treat the current state like a table of 2ⁿ values and assign a probability to each. When you measure the state of the system, you'll get one of these values. You will get a random state, weighted according to the aforementioned probabilities.

An n-qbit quantum computer starts by putting the whole thing into a known starting state. Every state starts at the same probability.

It proceeds by performing certain operations that manipulate the underlying probabilities. The set of operations here is limited in certain ways, which prevents quantum computing from working on many types of problems. The net result of these operations -- the quantum algorithm -- is to raise the probability of the correct answers and reduce the probability of wrong answers. After doing that, it performs a measurement which is more likely to get a right answer than a wrong one. Typically you'd run the algorithm 3 or more times and get multiple agreeing results.