failure is exactly that - when you cannot lift any more.
Say you are doing a bicep curl, if (for example) the 8th rep is very difficult, you get your arm all the way up to the top but feel tired and dont want to do any more - thats not failure!keep going! Do another rep - the point at which your muscle fails (you can no longer move the weight through its range any more) is when you stop.
On the last rep (where you fail and cant move any more), force youself to the end of the movement by cheating (using a spotter, helping with your other arm etc), but then leave it at that and rest.