I've been reading about base64 conversion, and what I understand is that the encoded version of the original data will be 133% of the original size.
Then, I'm reading about how YouTube is able to have unique identifiers to their videos like FJZQSHn7fc and the reason was: an 11 character base64 string can map to a huge number.
Wait, say a huge number contains 20 characters, then wouldn't a base64 encoded string be 133% of that size, not shorter?
I'm very confused. Are there different types of base64 conversion (string to base64 vs. decimal to base64), once resulting in a bigger, and the other in a smaller resulting string?
Think of it like this: you have a 64bit number (called long in Java, for example). 
Now, you can print that number in different ways:

As a binary number (base 2), printing 64 '0' or '1'
As a decimal number (base 10), printing up to 20 decimal digits
As a hexadecimal number (base 16), printing 16 hexadeciaml digits
As a number in base 64, printing 11 "digits" in that base. You can use any graphical symbols as digits.
... you understand by now that there are many more possibilities  ...

It seems like they use the same base-64 numbers as the ones that are used in base64 encoding, that is, uppercase and lowercase letters, ordinary digits and 2 extra chars. Each character represents a 6-bit value. So you get 66 bits, and depending on the algorithm used, either the leading or trailing 2 bits are cut off to get a nice long value back.
Base64 conversion decimals