A qubit represents a single bit, in its two possible states: 1 and 0. Alone, the qubit could hold the probabilities that the bit will be a 0 or a 1 when you observe it. But for a quantum computer to be cool, you have to include entanglement!
When 2 qubits are entangled, they represent the probabilities of each of the 4 possibilities: 00, 01, 10, 11. If you only had the probability of each bit separately, say 10%/90% and 40%/60%, then the state would just be the multiplication of the two values: 4% for 00, 6% for 01, 36% for 10, 54% for 11. And if the first bit's probability changes, the second bit doesn't change. A simple multiplication will give you the new answer.
But entanglement means the probabilities are dependent on the values of other bits :) So say: if the first is a 0, the second has a 90% chance of also being a 0. And if the first is 1, the second only has a 50% chance of being a 0 or a 1. So if the first has a 20% chance of being a 0, that gives us 18% for 00, 2% for 01, 40% for 10 and 40% for 11. The values of the bits are linked to each other. It's not limited to a simple linear percentage; it can get more complicated by running the bits through various gates. So if the first bit's probability changes, you have to recalculate the new probabilities of B given A. That's when it would take exponential time using a classical computer, but it's all done instantaneously via entanglement.