autofocus - What does infinity focus mean and when should I use it?

All my lens have focus ring where at the end there's an infinity symbol. What does that mean? If I'm taking landscape shot and what everything from near to far to be in focus, do I use infinity? Or am I better of focusing onto a single point one third up from the bottom (I've read that, that's the rule of thumb).

Also, why is infinity a range on most lens. It's never a single point.