Each base follows a pattern and you can clearly see that from the examples.

Hexadecimal (Base 16)#

Each digit can represent upto 16, 0-9 and then A-F(for 10 to 16).

0x2f is 48 in decimal

    2                f
(16^1)*2  + (16^0)*16(f = 16)

Decimal (Base 10)#

The one we’re all familiar with, numbers 0-9. Pretty self explanatory.

25 

    2           5
(10^1)*2  + (10^0)*5

Octal (Base 8)#

Contains numbers 0-7, each digit representing 3 bits. If you’re familiar with Linux file permissions you already know Octal.

10 in decimal would be represented as 12 in octal 

  1           2
(8^1)*1  + (8^0)*2

Binary (Base 2)#

Contains just 0 and 1, really useful when working with circuits and can represent a on/off state.

101 in binary is 5 in decimal

   1         0        1
(2^2)*1 + (2^1)*0 + (2^0)*1

Why so many?#

Binary is most useful when dealing with computers as circuits can only represent 2 states at a time, one being on and another being off.

You might occasionally see Octal being used, the best example I can think of is file permissions in Linux.

Hexadecimal is very extensively used to represent memory or values in a system as these values can get very huge.

For example 64GB in bytes is 68,719,476,736. In hexadecmial? 0x1000000000. As you can see it is much easier to read and represent hexadecimal numbers once you get used to them.