Binary is counting in two. Instead of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, we simple count 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001 and 1010. This reduction of counting to the use of two symbols is both imagative and very powerful because it greatly faciltates the mechanisation of countng. For example imagine we have four egg cups sitting in a row on a bench. We can represent any number from 0 to 15 simple by deciding that an upside down egg cup represents "1"; so if the first two egg cups are upside down and the two remain right side, this translates to "1100" in binary and "12" in base 10.
Other mathematical operations are also simple in binary, for example, adding in binary is quite elegant, for example think of 5 + 7 perfomed in binary :
101+ 111 = 1100
This whole procedure can be reduced to a simple recipe. To get the idea, line up two rows of four egg cups (assuming your family like boiled eggs alot or you have a big family!!) and see whether you can devise a set of rules for add two numbers together. Try to do the same with eight egg cups and adding decimal numbers together .... now you can start to appreciate the power of binary.
The binary number system combined with Boolean logic (another great feat of mathematical imagination) is central to the workings of the modern computer. Counting, number manipulation and storage, can be performed with amazing speed and accuracy based on very similiar procedures to your egg cup algorithm.