When this creature dies, if it had no +1/+1 counters on it, return it to the battlefield under its owner's control with a +1/+1 counter on it.
Many players have heard of persist, an ability in the Shadowmoor Block. Persist was very similar to undying except that it dealt with -1/-1 counters instead of +1/+1 counters. Basically, a creature with undying is like a normal creature until it dies, and when it does, it will come straight back to the battlefield. Note that is says that it will only come back if it didn't have any +1/+1 counters on it. That means that you can only use it once, unless you could remove the counter, and also it will not come back at all the first time if you use something like Soul's Might to put +1/+1 counters on it.
Flavorwise, undying is meant to be a way to get the feel of living beings dying and coming back after death as zombies , geists, or vampires. It was returned in Avacyn Restored as a feeling that evil was not entirely gone.