Binary Digits
A Binary Digit can only be 0 or 1 
Binary NumberA Binary Number is made up Binary Digits. 
Bits: In the computer world "binary digit" is often shortened to the word "bit"
More Than One Digit
So, there are only two ways you can have a binary digit ("0" and "1", or "On" and "Off") ... but what about 2 or more binary digits?
Let's write them all down, starting with 1 digit (you can test it yourself using the switches):
2 ways to have one digit ... 


... 4 ways to have two digits ... 


... 8 ways to have three digits ... 


... and 16 ways to have four digits. 

And, in fact, we have created the first 16 binary numbers:
Decimal:  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 

Binary:  0  1  10  11  100  101  110  111  1000  1001  1010  1011  1100  1101  1110  1111 
This is quite a useful thing to remember. If you forget how the sequence of binary numbers go, just think:
 "0" and "1" {0,1}
 then "0" and "1" again but with a "1" in front: {0,1,10,11}
 then take those four with "1"s as a third digit: {0,1,10,11,100,101,110,111}
 and so on!
(Or count in the same way as decimal numbers, but don't use 2,3,4,5,6,7,8 or 9.)
Have a Play with 4 different drums. 
Binary Digits ... They Double!
Also notice that each time you add another binary digit you double the possibilities. Why double? Because you have to take all the previous possible positions and match them with a "0" and a "1" like above.
So, if you had 5 things, then the total would be 32, 6 things would be 64, etc.
Using exponents, this can be shown as:
No of Digits  Formula  Settings 

1  2^{1}  2 
2  2^{2}  4 
3  2^{3}  8 
4  2^{4}  16 
5  2^{5}  32 
6  2^{6}  64 
etc...  etc.^{..}  etc... 
Example: when you have 50 binary digits (or even 50 things that can only have two positions each), how many different ways is that?
Answer 2^{50} = 2 × 2 × 2 × 2 ... (fifty of these) = 1,125,899,906,842,624
So, a binary number with 50 digits could have 1,125,899,906,842,624 different values.
Or to put it another way, it could show a number up to 1,125,899,906,842,623 (note: this is one less than the total number of values, because one of the values is 0).
Chess Board
There is an old Indian legend about a King who was challenged to a game of chess by a visiting Sage. The King asked "what will be the prize if you win?".
The Sage said he would simply like some grains of rice: one on the first square, 2 on the second, 4 on the third and so on, doubling on each square. The King was surprised by this humble request.
Well, the Sage won, so how many grains of rice should he receive?
On the first square: 1 grain, on the second square: 2 grains (for a total of 3) and so on like this:
Square  Grains  Total 

1  1^{}  1 
2  2^{}  3 
3  4^{}  7 
4  8^{}  15 
10  512^{}  1,027 
20  524,288 
1,048,575 
30  53,6870,912 
1,073,741,823 
64  ??? 
??? 
By the 30th square you can see it is already a lot of rice! A billion grains of rice is about 25 tonnes (1,000 grains is about 25g  I weighed some!).
Notice that the Total of any square is 1 less than the Grains on the next square (Example: square 3's total is 7, and square 4 has 8 grains). So the total of all squares is a formula: 2^{n}1, where n is the number of the square. For example, for square 3, the total is 2^{3}1 = 81 = 7
So, to fill all 64 squares in a chess board would need 2^{64}1 = 18,446,744,073,709,551,615 grains (460 billion tonnes of rice), many times more rice than in the whole kingdom.
So, the power of binary doubling is nothing to be taken lightly (460 billion tonnes is not light!)
(By the way, in the legend the Sage reveals himself to be Lord Krishna and tells the King that he doesn't have to pay the debt immediately but can pay him over time, just serve rice to pilgrims every day until the debt is paid off.)
Hexadecimal
Lastly, I would like to tell you about the special relationship between Binary and Hexadecimal.
There are 16 Hexadecimal digits, and we already know that 4 binary digits have 16 possible values. Well, this is exactly how they relate to each other:
Binary:  0  1  10  11  100  101  110  111  1000  1001  1010  1011  1100  1101  1110  1111 

Hexadecimal:  0  1  2  3  4  5  6  7  8  9  A  B  C  D  E  F 
So, when people use computers (which prefer binary numbers), it is a lot easier to use the single hexadecimal digit rather than 4 binary digits.
For example, the binary number "100110110100" is "9B4" in hexadecimal. I know which I would prefer to write!