Binary Digits
A Binary Digit can only be 0 or 1 
Binary NumberA Binary Number is made up Binary Digits. 
In the computer world "binary digit" is often shortened to the word "bit"
More Than One Digit
So, there are only two ways we 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. 

Here is that last list sideways:
0000  0001  0010  0011  0100  0101  0110  0111  1000  1001  1010  1011  1100  1101  1110  1111 
And (without the leading 0s) we have the first 16 binary numbers:
Binary:  0  1  10  11  100  101  110  111  1000  1001  1010  1011  1100  1101  1110  1111 

Decimal:  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 
This is useful! To remember the sequence of binary numbers just think:
 "0" and "1" {0,1}
 then repeat "0" and "1" again but with a "1" in front: {0,1,10,11}
 then repeat those with a "1" in front: {0,1,10,11,100,101,110,111}
 and so on!
At each stage we repeat everything we have so far, but with a 1 in front.
Now find out how to use Binary to count past 1,000 on your fingers:
Also have a Play with different drums. 
Binary Digits ... They Double!
Also notice that each time we add another binary digit we double the possible values.
Why double? Because we take all the previous possible values and match them with a "0" and a "1" like above.
 So just one binary digit has 2 possible values (0 and 1)
 Two binary digits have 4 possible values (0, 1, 10, 11)
 Three have 8 possible values
 Four have 16 possible values
 Five have 32 possible values
 Six have 64 possible values
 etc.
Using exponents, this can be shown as:
Number 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 we have 50 binary digits (or 50 things that can only have two positions each), how many different ways is that?
Answer: 2^{50} = 2 × 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).
Example: Start the month with $1 and double it every day, after 30 days you will be a billionaire!
2^{30} = 2 × 2 × 2 × 2 ... (thirty of these)
= 1,073,741,824
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 is 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 = 8−1 = 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!)
Grains of rice on each square using scientific notation
Values are rounded off, so 53,6870,912 is shown as just 5×10^{8}
which means a 5 followed by 8 zeros
(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 at once, but can pay him over time, just serve rice to pilgrims every day until the debt is paid off.)
Hexadecimal
Lastly, let us look at 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!