here's a short definition from a computer engineer buddy of mine (from an online conversation):
David: Suuuper simple
They are a thought experiment.
And are useful for proving things (ie, what a computer can and can't do)
David: basically a turing machine is a black box and a piece of tape
and the tape has writing on it
the box can read one character of writing at a time, then either move the tape left or right, replace the character it's reading, or do nothing
and that's pretty much it
but turing showed that that's all you need to do every computation a modern computer can do
David: so modern computers are modeled after the box
me: why is the infinite tape necessary?
David: there are a whole bunch of computations that you can't do with a turning machine
David: but the "tape" in a computer is 8.79609302 × 10^12 characters long, so essentially it’s the same as a Turing machinefor a more in-depth description from another computer scientist (who doesn't believe modern computers qualify as Turing machines!) click here