Prime numbers
What are they and why should we care about them? Well, they are the fundamental building blocks of all other numbers.
Brent Yorgey found what I think is perhaps the best way of displaying prime numbers: factor diagrams. Now we all know a prime number is a number that is only divisible by itself and one, but what does this really mean? It's really not all that intuitive.
A better way to think about prime numbers is that they are the numbers which can't be grouped into groups of the same size. Let's take a non-prime number, 12, as an example. This can be grouped into 2 groups of the same size, each group being of size 6. This can be done recursively, 6 can be grouped into 2 groups of size 3. So 12 can be grouped into 2 groups of 2 groups of size 3.
The animated gif above shows the factor diagrams for the numbers 10 through 20. Notice that the prime numbers all form rings. This is because these are the numbers that cannot be grouped into subgroups that have the same size!
In my spare time a few years ago I made a little program which smoothly animates the factor diagrams for all numbers. Here is the link for your viewing pleasure. Careful! You may end up hypnotising yourself.