Rooting is the act of getting admin access on your phone - just like the Administrator access on your pc.
It allows you to do a number of otherwise prohibited or poorly implemented things -
- full backup
- ad blocking
- proper firewall
- custom operating system installations or limited patches from the development community to control theming, efficiency, security, control, updates not according to HTC or the carrier
It's not for everyone but those who succeed at it typically don't go back to a non-rooted phone again.
Very, very few apps refuse to work on a rooted device. I've yet to have one but I've read about a couple - literally, two or three.