A binary number is a number expressed in the base of 2. Only two digits are used: 0 and 1.
Counting in binary from 1 we have 1, 10, 11, 100, 101, 110, 111, 1000, etc.
1+1=10 because we have no digit 2 in this system. 10 means 1×2+0; 101 means 1×4+1 (4 is 22=100 in binary).
Imagine you had a row of cups. When they're all empty this represents zero.
Now fill the end cup. This represents 1, because all but one cup is empty.
If we try to add another cupful to the end cup it would overflow, so to make room we must empty the cup into the one next to it. So now we have an empty cup at the end but the one next to it is full. That completes the addition and represents the binary number for 2 which is 10: a full cup next to an empty cup.
If we add another cupful we can do this by simply filling the last cup, and this represents binary for 3: 11, two full cups.
To add another cupful to the two already full we have to fill the third cup from the end because: empty cup 2 into cup 3, then empty cup 1 into cup 2 leaving cup 1 empty. That completes the operation of adding 1, so we end up with 100, binary 4. Binary 5 would allow us to fill cup 1: 101.
Adding another cupful would, by overflow fill cup 2: 110, binary 6. Binary 7 is 111 because we can fill cup 1.
Each time we add 1 we either cause at least one overflow or we can fill the last cup. Each cup in turn represents a power of 2: 1, 2, 4, 8, 16, 32, etc.
Doubling in binary simply means we move the contents of the cups up one, which is the same as adding a zero to the end of the binary number, just as multiplying by ten in decimal adds a zero onto the end (or moves the decimal point). So, for example, 11 (binary for 3), 110 (binary for 6), 1100 (binary for 12), 11000 (binary for 24), etc.