If you look at the set of real numbers, if you take a range of real numbers (for example all the real numbers between 1 and 2), that set will still be infinite (since there are an infinite amount of numbers between one and two. So real numbers are uncountably infinite.
If you look at the set of integers (1,2, 3, 4....). A range of numbers in that set will actually only be a finite subset (for example, the range between 3 and 5 will only contain 3 elements. So, integers are countably infinite.
There is actually more to the definition, but this is the simplest explanation that makes sense to me.
Thanks for trying to explain it to me, but unfortunately (for me) I'm still confused. It seems to me that the set of real numbers between 1 and 2 is infinite only because the number of digits to the right of the decimal point can be increased arbitrarily so that the set is countably infinite in terms of the number of digits required to represent it. I guess I am assuming (perhaps wrongly) that if an infinite set is composed of a number of countably infinite sets it itself is countably infinite. (See, I told you this wraps me around the axel.) And, of course, my thinking could be totally wrong. If you feel up to it, please teach me more.