Binary Counting ("Base 2")
Computers do not count with the same counting system we use. Humans have ten fingers, so we use a base10 counting systemthat is, 10 digits. A "digit" is a singlecharacter number; the ten digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Computers use switches to keep information. A switch can be off or on. On is 1; off is 0. SO computers only use those two digits. One numbera 1 or a 0is called a binary digit, or a "bit" for short.
When we count, we start with a single digit, going from 0 to 9. After 9, there are no more digits. So we have to use two digits. Each digit represents a "column," as shown in the chart below. In the number "10," the "1" is in the "tens" column, and equals "1 x 10"; the "0" is in the "ones" column, and equals "0 x 1."
As an example, take the number 256:
Start at the right, and move to the left

Millions 
Hundred Thousands 
Ten Thousands 
Thousands 
Hundreds 
Tens 
Ones 
1,000,000 
100,000 
10,000 
1,000 
100 
10 
1 
10^{6} 
10^{5} 
10^{4} 
10^{3} 
10^{2} 
10^{1} 

0 
0 
0 
0 
0 
0 
0 
1 
1 
1 
1 
1 
1 
1 
2 
2 
2 
2 
2 
2 
2 
3 
3 
3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
4 
5 
5 
5 
5 
5 
5 
5 
6 
6 
6 
6 
6 
6 
6 
7 
7 
7 
7 
7 
7 
7 
8 
8 
8 
8 
8 
8 
8 
9 
9 
9 
9 
9 
9 
9 
Therefore, the number "256" means that you have 2 hundreds, 5 tens, and 6 ones. Each digit has ten possibilities, so with each new digit, you multiply by ten. A 2digit number has 100 possibile combinations. A 3digit number has 1000 combinations, and so on.
However, if you only have two numbers instead of ten, your counting has to look like this:
Start at the right, and move to the left

five hundred and twelves 
two hundred and fifty sixes 
one hundred and twenty eights 
sixty fours 
thirty twos 
sixteens 
eights 
fours 
twos 
ones 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2}

2^{1} 

0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
Notice the numbers: 8, 16, 32, 64, 128, 256, 512. These are numbers seen very often with computer memory.
To write the number 7, you would write "111"that is, 1 four, 1 two, and 1 one. To write the number 8, you would write "1000"1 eight, and 0 fours, twos and ones. The number 256, therefore, is "10000000."
Notice that the second column is the "base," and all other columns are powers of the basebase^{2}, base^{3}, base^{4}, etc.
Here is an example of counting from one to ten in binary:
binary 

base 10 
0 

0 
1 

1 
10 

2 
11 

3 
100 

4 
101 

5 
110 

6 
111 

7 
1000 

8 
1001 

9 
1010 

10 
Each digit has two possibilities, so with each new digit, you multiply by two. For example, a 2digit binary number has four possible combinations (00,01,10, and 11). A 3digit number has eight combinations; a 4digit number has 16 combinations, and so on.
Try using this binary  base 10 conversion app to see how binary numbering works:
