How does BitCoin Mining work?

New to the BitCoin economy. But I don't exactly understand the concept of it.
I understand that I need a computer to "mine" bit coins but why is a extremely powerful computer necessary to solve the "blocks"
Are the problems mathematical?

And what does a HASH measure or represent. I saw that some Bitcoins take over a year to mine on slow computers but what does that mean...?

I'm sure these are basic things but I can't seem to find a concise answer online. This new this idea seems fascinating and I want to get involved. Thanks any and all help is appreciated.