There are actually three types of failure (that I'm aware of) and happen in this order during exercise:
Concentric failure - you can no longer complete the positive (concentric) part of the lift without sacrificing form.
Static failure - you can no longer hold the weight still.
Eccentric failure (AKA total/complete muscular failure) - you can no longer control the weight on the negative (eccentric) part of the lift.
I prefer training to concentric failure, but everyone once in a while static failure. Eccentric failure is rather painful and usually requires help from a spotter...