[summary: Binary notation is a way to write numbers using [math_power multiples] of two instead of the usual multiples of ten. It's especially useful for working with computers, which use binary notation exclusively.]
When you were first taught how to write numbers and [addition add] them together, you were probably told something about the "ones place", the "tens place", the "hundreds place", and so on. Each [-digit] further to the left represents a larger [math_power multiple] of ten, and the multiples are added together to get the number you want--so $~$8207$~$ is counted as $~$(7 \times 10^0) + (0 \times 10^1) + (2 \times 10^2) + (8 \times 10^3)$~$, or "seven ones, zero tens, two hundreds, and eight thousands". But why use powers of ten? What's so special about that number?
Well, nothing, actually. We probably only use powers of ten because we happen to have evolved with ten fingers--any number bigger than one will work just as well, although it may look strange at first. Since two is the smallest Integer larger than one, a binary number notation--a notation that uses powers of two--is one of the simplest possible. %note:The word "binary" has the same root as the word "bicycle"--bi meaning "two".%
Binary notation uses only two digits, $~$0$~$ and $~$1$~$, and each "place" to the left goes up by a power of two instead of ten (in other words, it doubles). For example, in binary notation, the number $~$11010$~$ is counted as $~$(0 \times 2^0) + (1 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4)$~$, or "zero ones, one two, zero fours, one eight, and one sixteen". Translating back to the familiar [number_bases base] ten, we would write it as $~$26$~$.
You may notice that binary notation tends to be a bit longer than decimal (for example, "11010" takes more characters to write than "26"). It's also more difficult to read, unless you have a lot of practice with it. So why would anyone use it? Well, for one thing, binary notation is often very convenient for talking about powers of two, for instance when working with base-two logarithms or log Odds, or when working with some quantity measured in bits. It's also essential for working with computers, as all modern computers store and manipulate data, on the lowest level, exclusively using binary notation.