Defined as pushing past the range of motion barrier to where you can no longer move the weight in the desired exercise even with some changes in tempo or loss of range of motion. There is grey area into how much loss of execution is allotted for, but the only acceptable complete failure methods I suggest are loss of range of motion, and maybe a minimal amount of tempo or recruitment.
In other words, if you are doing a set of bicep curls, it is better to force some reps where you can not get the weight all the way up and just go back down and do a few more controlled partial reps than it is to swing the weight from the bottom and then drop it fast. Losing some range of motion but still contracting the working muscle, and controlling it allows you to fatigue that working muscle further. Adding your hips and back into a bicep curl may increase the amount of mechanical damage due to more eccentric work but it also increases your risk of injury.
When programming for clients we will sometimes define a percentage range of motion failure. For example, a 50% ROM failure on a leg extension would be when you can no longer get 1/2 way through the concentric portion of the rep.
This can be used in most metabolic and hypertrophy phases, depending on the stimulus you’re after and how good the execution of the individual is.