Repeater serve 3 functions: A delay circuit, a diode, and a signal extender.
Delay Circuit[]
Repeaters have 4 modes they can be set to, each correlating with how much to delay the signal. By right clicking on the repeater, the delay increased. The 4 delay times are: 0.1 second, 0.2 second, 0.3 second, and 0.4 second. When the repeater is right clicked when at 0.4 delay, it will go back to 0.1. Delays are helpful in the timing of circuits and the generation of clock pulses. The delay is also what makes Scrolling Text Displays possible, since the signal cascades down the line of repeaters.
Diode[]
With repeaters, the signal can only go one way, making it a diode. This is useful for circuits where one signal could interferewith another signal. Also, it makes it easier to cross signals, since the redstone will not try to bond with the repeater if commig from the side.
Signal Extender[]
Before the addition of repeaters, signals that needed to be carried for more than 7 blocks had to have a double NOT gate to extend the signal. Repeaters can extend the signal 15 blocks before needing to be extended again. They are a lot more compact than the double NOT gate, and faster, since the minimum repeater delay is .1 second, while the double NOT gate is around .2 seconds.