"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.