The main purpose of the tripod is to keep the camera absolutely still so as not to introduce camera shake. Used mainly for macro photography, portraits, night shots )moon, stars, etc.).
For older guys like me that can't hold their cameras steady! lol It's amazing how much movement the mirror operation shakes a camera on close up macro or long exposure shots.
Various reasons, as already mentioned, plus for taking night shots where the camera is in the same place for a long time, doing multiple exposures, or as already stated, to hold the camera still so you don't get camera shake for old people like me.
There as more uses for a tripod then I can think of. I bet others can think of more, and I am sure more will be thought of in the future.
One word...stability. Rules for getting the sharpest images possible:
Use a tripod.
Use a remote or the timer function.
Use both whenever possible.
Even the slightest bump (especially at high zoom or macro) can remove sharpness from your photographs. Something as simple as manually pressing the shutter button causes movement.
Obviously it's not always ideal to use a tripod, but you can in most situations.