The book does a nice job on this, especially for integer arithmetic; read it. You should be aware of 1's complement, 2's complement, and sign + magnitude integer formats; integer addition, negation, and subtraction algorithms and hardware (ripple, lookahead, and select carry processing); and the integer multiply and divide discussed in class and the text. You should also have a basic understanding of floating point.
In discussing integers, we also briefly discussed BCD (Binary Coded Decimal) and Gray codes (efficient conversion between 2's complement and Gray codes is given here). We also mentioned saturation arithmetic (as opposed to the modular integer arithmetic most commonly used). However, the primary addition to the book material on integer arithmetic was our discussion of speculative addition. Our in-class discussion was actually a slightly simplified version of what Intel patented. Here is a nice little article discussing what Intel implemented in the Pentium 4.
It also is worth noting that floating point arithmetic differs from real number arithmetic in many important ways, which the book doesn't really cover very well. I don't expect you to understand the subtle nuances of floating point, but you should be aware that it is a very strange beast. Toward that goal, you might browse through What Every Computer Scientist Should Know About Floating-Point Arithmetic, which details many of the stranger quirks of floating point arithmetic. Don't read this reference too carefully; it is overkill for the purposes of this course. A lighter, but similar, document is The Perils of Floating Point, which is quite readable despite talking about Fortran. Most fundamentally, you should be aware of the accuraccy loss issues involving addition and subtraction and the IEEE base 2 floating point format.
Of course, THE standard for floating point is IEEE 754 (and, like most IEEE standards, it isn't free to get a copy of the standard itself ;-).
Computer Organization and Design.