A set is uncountable if it contains so many elements that they cannot be put in one-to-one correspondence with the set of natural numbers. In other words, there is no way that one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.
Uncountable is in contrast to countably infinite or countable.
For example, the set of real numbers in the interval $[0,1]$ is uncountable. There are a continuum of numbers in that interval, and that is too many to be put in a one-to-one correspondence with the natural numbers. One can show using Cantor's diagonal argument that for any infinite list of numbers in the interval $[0,1]$, there will always be numbers in $[0,1]$ that are not on the list.