A quantum computer is just a computer which uses quantum states instead of classical states. Quantum states are just combinations of classical states.
Say a computer takes a certain input $x_i$ to output $f(x_i)$. According to quantum mechanics, you can set up your computer such that its input is a combination of all of the classical inputs $x_i$, namely $\frac{1}{n} \sum_i^n x_i$. After operating, your computer would then take this combination of all inputs to the combination of all outputs $\frac{1}{n} \sum_i^n f(x_i)$.
If this is all there was to it, we could compute all of the outputs of a function as fast as we could compute one. The quantum caveat is that when you go to look at your answer, you don't see $\frac{1}{n} \sum_i^n f(x_i)$, you see $f(x_i)$, for some $i$, with probability $\frac{1}{n}$. So it looks like we've only computed the function once after all! However, if we're clever, we can still benefit from our quantum computer. Let's say, instead of wanting to know all the values, $f(x_i)$, we want to know some function $g(f)$, which depends on all of the answers $f(x_i)$. If we're clever enough, we can sometimes come up with an algorithm which takes $\frac{1}{n}\sum_i^n f(x_i)$ to $g(f)$. The classical computer might have needed to compute every $f(x_i)$ just to find $g$, but the quantum computer can just do it "once."
Please Note: this isn't exactly how it works in most cases, but most cases require quite a bit of math to describe accurately. But this is the "general idea" of why quantum computers can outperform classical ones.
