A German mathematician called Leopold Kronecker once said that "God gave us the integers, and all the rest is the work of man." I think what he was getting at is that numbers exist, but we can choose to represent them in different ways, and do different things with them. Without getting too philosophical, it helps to remember that, often, what we think of as numbers are actually symbols representing the number - just as the way you write your name isn't you, and you're still the same person if you type your name, write it in a different colour, or use a different alphabet.
Number bases are different ways of writing and using the same number. We use a system called base 10, or denary, for our arithmetic, but there are almost as many number bases as there are numbers. Many people think that we use base 10 because we have 10 fingers on which we can count. Computers, and other electronic devices, can only reliably use an electrical current, or the absence of a current, to count (like having two fingers), and so they tend to use base 2 (binary) internally.
You can check out more about it here.