ASCII is the decimal (base 10 (0 through 9)) version of each keystroke. For example, the letter 'A' is represented by the ascii value of 65. You can use the mirc $asc to show this.
Binary is base 2 (0 and 1).
Hex is base 16 (0 through 9 then A through F). The easiest way to work out the hex of a number is to go through binary and splitting into 2.
E.g.For the capital letter A:
ASCII = 65
Binary = 01000001
Now split the binary into 2 to get 0100 and 0001. Convert these to to decimal numbers, you get 4 and 1. So the hex value for A is 41.
ASCII generally only goes upto about 128 or something like that, so is very limited, whereas Unicode uses 10,000's so can be used for different languages with special characters.
A checksum is usually a calculation, depending on the checksum used, depends on what the calculation is. This calculation is done prior to transfering, and after as a means of checking the bits (a single binary digit) are all in the right order. This could be as simple as 'hex + 1' but that wouldn't be a very accurate method.
The first part of what you pasted is the 'Unicode Hex' values for the letter stings in brackets. If is its showing a checksum, then the second part is the result of the letters a, b, c, aaaa etc after the checksum is done.
Hope this helps