If a Saiya-Jin does not have a tail, their powers aren't as strong as they could be. Tails do regrow, but this takes some time. The tail is used for transforming into their more powerful form.
A Saiya-Jin's tail is a very sensitive spot, so whenever an enemy pulls on a tail, the Saiya-Jin is rendered completely immobile. Only the absolute elite fighters of the first class have managed to train hard enough to overcome this weakness.