The simplest way to represent numbers using circuits is to only distinguish two values: 1 and 0, or "on" and "off".
There are many physical ways of representing these two values, including:
We are used to represent numbers in the decimal system, where we have 10 number symbols: the digits 0..9.
If we only have two digits 0, 1, we have to use the binary system. Each digit is called on bit; and 8 bits form one byte.
An explanation of the binary sytem as well as binary arithmetic can be found here.
An additional short explanation of using "Two's Complement" to represent negative numbers can be found here.
Personally, I think a binary odometer provides the best motivation for representating negative numbers using "two's complement", but not very many people have the luxury of owning one. Here is one explanation that uses the binary odometer anlogy (search for "odometer" on this page).
For the exam, I'm only expecting you to be able to convert positive numbers between decimal and binary, and to do some simple binary arithmetic (addition).