What is the difference between countable infinity and uncountable infinity?

Answer:

First we need to recognize that any form of infinity is just a mathematical construct. We are not ever going to have to deal with actual infinities, but what is very useful is understanding limiting behavior as something or other goes towards infinity. It is here that the difference between countable and uncountable infinities matters immensely.

countable

can be put into a one to one correspondence with the natural numbers

uncountable

cannot be put into a one to one correspondence with the natural numbers