When lifting, is failure being able to complete your last rep but struggling to do so? Or is it not being able to complete that last rep?

For you C25K people, on the last day of the week are you completing that run before moving on to the next week? Ex: being able to run the full time of 3 min runs on week 3 before moving to the 4 and 6 min runs of week 4?

Failure is failure. Can not complete.

Yup - failure is getting 1/2 way through a lift and "failing" to finish it. I go to failure all the time when I am testing my 1 rep or 3 rep of the standard lifts. For me, the easiest way to failure is on the overhead press, no jerk. Anywhere from 2 1/2 lb to 5 lb increase and I will bonk out, hitting a wall. I get my arms 1/2 way up and just cannot straighten them. That is failure. The good thing about failure is it teaches your muscles to grow and get stronger. For example, my overhead press (no jerk) is 150 currently. It has taken me a couple months to go from 85 to 150 because of bonking at that wall. For dead lift, bench press and back squat, I am much stronger. I can increase more without bonking. However, I do need to always be aware of injury so do not lift into a zone of injury, especially if the weight is getting to be really heavy, like 325 on dead lift. At least I can drop at the top of a 1 rep dead lift. For 3 rep dead lift, I need to lift a little less because good form needs to be maintained through all 3 reps. Same is true if I am doing a 10 rep set. Do not lift so much you injure yourself on the way to failure.