Divide numbers -- or sets of things -- into two groups: finite and transfinite. Transfinite numbers -- or sets -- can further be divided up into two kinds: countable and uncountable. Finite sets are those you that you can actually assign a number to the amount of elements in the set. Put another way, if you have an algorithm for listing elements of a finite set, the algorithm will end at some point. Another characteristic of finite sets is that if you compare a finite set with a second set made by taking a few elements out, the second set will have fewer elements. These are not characteristics of transfinite sets.
Describe countable sets as sets of things you can start counting but the counting never stops. An example is the positive integers: 1, 2, 3, 4, 5, and so on. One of the characteristics of countably transfinite numbers sets is you can put the elements of the sets in a one-to-one correspondence with a proper subset of itself. For example, the integers can be put into a one-to-one correspondence with the even integers. Each integer corresponds to an even integer: twice the integer, and each even integer corresponds to a unique integer: half the even integer. This could never happen with finite sets.
See that there must be another kind of infinity that cannot be put into a one-to-one correspondence with a countably infinite set. The model for an uncountably infinite set is the real numbers. There is a "next element" after the integer 6, but there is no "next element" after the real number 6. To see the difference between countably infinite and uncountably infinite, consider points and lines. Points have no length or width, but lines are made up of points. You could never put together enough points to make a line. There must be some kind of infinity that is more than "start counting and keep going."