The most simple answer I can come up with is this: "classical computers" (as they are now known) work with 0's and 1's only, which can be thought of as "yes" and "no" only. Which in turn makes them great at anything with definite in inputs and leads to everything computers can do today. The problem is, it makes them very bad at anything that needs to deal with "maybe" and "probably" at all.

Quantum computers in contrast, work exclusively with "maybe" and "probably". Which means things like "true AI" (like C-3PO) will be possible. Weather forecasting will get MUCH better. Machines won't be limited to doing "exactly what you said"... they will be able to "do what you meant". Anything having to do with "probability" instead of "certainty" (which is currently nearly impossible to work with) will suddenly be as easy as using Excel to record item prices and produce an average.

In addition to all of that, quantum computes work with much more information at a time. Again, keeping it very simple: classical computers work with individual bits to make a byte which represents (roughly) a single letter or number; quantum computers can work with entire concepts at a time.

All of this is also why "what kind of solutions will it excel at?" is a really hard question. It's like trying to come up with answers about what the internet will be good for... in 1910 or so.