"Pettigrew owes his life to you. You have sent Voldemort a deputy who is in your debt... When one wizard saves another wizard's life, it creates a certain bond between them... and I'm much mistaken if Voldemort wants his servant in the debt of Harry Potter... This is magic at its deepest, its most impenetrable, Harry."
A life debt is a magical bond formed between a wizard or witch and the person whose life they saved. The one who owes the debt to the saviour would one day be obliged to repay the deed by doing something beneficial to the said saviour. Such a bond can be formed between even the worst of enemies and regardless of whether either of the involved parties want it or not. As this bond is magically binding, the indebted could commit the repayment without acknowledging it, or even against their will.
In his third year Harry prevented Peter Pettigrew from being murdered by his former best friends, Sirius Black and Remus Lupin, for treason. Instead, Harry asked for him to be transferred to Azkaban. Though Pettigrew escaped, he still owed Harry a life debt. Perhaps due to gratitude, Peter suggested to Voldemort on using another wizard for the potion to recreate his body. This was repaid when Pettigrew showed a rare and brief moment of mercy at the Skirmish at Malfoy Manor and released Harry from a choking hold with his silver hand, which Voldemort had given to him for sacrificing his own hand to bring Voldemort back into human form. Unfortunately, it was fulfilling this debt that caused the silver hand to then turn on Peter, killing him for showing what Voldemort would have considered weakness.
Behind the scenes
Both known life debts are related to Harry's benefit, and both the indebted parties died protecting him.
It is unknown whether Arthur or Ron ever had life debts to Harry, for being rescued after being attacked by Nagini in 1995, and given a bezoar after being poisoned in 1997, respectively. If it is true that Ron once owed Harry a life debt, he presumably repaid it by saving Harry from drowning in 1998. Seeing that Ginny did not owe Harry a debt, it may be possible that neither Arthur nor Ron owed him either.
It is unknown if life debts can be made between animals and people, but if so Harry may be in debt to Fawkes the Phoenix for saving him from the Basilisk venom, and Dumbledore might have been in debt to Fawkes for swallowing the Killing Curse during his duel with Voldemort, although it should be noted that an instance where a Phoenix would require the repaying of a life debt would be rare, if even possible, given that they are, in all known cases, reborn after death.