Quantum computing can solve certain problems exponentially faster, but not all.
A classical computer with $n$ bits can store a single number with value up to $2^n$, while a quantum computer with $n$ bits can store a superposition of all $2^n$ basis states (i.e. a superposition of all the classical inputs). You can then run whatever operations you want on that large input state to end up with a superposition of that algorithm ran simultaneously on each classical input.
However, extracting all of that information is impossible. While the quantum resource of superposition lets you evaluate on every classical input at once, naively measuring the resulting quantum state will collapse it to the single output corresponding to one of your classical inputs. You may not even be able to trace the output you got back to the classical input which generated it. Also, if you want to know the output of your algorithm on a single classical input (or for every classical input), you can do that as efficiently on a classical computer as you can on a quantum one.
Where quantum computing provides a massive benefit is where there's some symmetry or periodicity that can be used to make undesired outputs cancel each other out. For example, Shor's algorithm takes a superposition of classical inputs $\{x\}$, evaluates $f(x) = a^x \mod N$ on each $x$ (where $a$ is random and $N$ is a number we'd like to factor), then applies a Quantum Fourier Transform which destructively interferes the undesired outputs and constructively interferes the correct ones so that when you do make your single measurement before collapsing the state, you're very likely to get a useful answer.
In general, efficient quantum algorithms have some clever step that whittles down a large superposition to a state that lets you make the most of your single measurement, and the speed up you get from quantum computing depends on whether there's such a step that works on your problem.
