More than just holding the reed on - though that's essential. The ligature also impacts the degree to which the reed is able to vibrate. A heavy and tight ligature reduces the vibration and results in a tone that is more "woody" but that doesn't respond as fast when you blow. (it takes more air)
A loose and light ligature lets the reed vibrate freely which makes it answer very fast when you blow but also tends to make it over-answer (SQUEEK!) if you're not careful.
There are all sorts of ligatures out there that range from light to heavy materials and tight to loose holding power and any combination in between. Rovner is an example of good aftermarket ligatures especially popular with jazz players.
The simplest ligature (light and loose) is to wrap string around the reed to hold it on. It's called string-wrap and is what was used by almost everyone before the 1920's. Vandoren makes one called the Klassik that is as close to a string wrap as you can get without the inconvenience of wrapping string.
Your choice depends mostly on your own preferences though and the only way to find out what you like best is to try them on your instrument. A quality ligature is a pretty inexpensive way to make a major improvement to your instrument.
So, the real purpose of the ligature is to control the vibration of the reed. If it was only to hold the reed on then we'd use screws, glue, or tape.