"Mean time to failure" is just what it says: the "Time to Failure" is the length of time from putting something into service (a new automobile, a tooth implant, quitting smoking) until that something fails (the car breaks down, the implant falls out, you light up again). Try this a bunch of times, putting identical things into service all under the same conditions and recording the times to failure, and then take the "mean" of those times (you may know "mean" as "average": add the times to failure up and divide by how many were tested). That's the "mean time to failure". The trick is to make the things and the their service conditions "identical", and to test enough of the things to get good statistics. But that's another question.