What is a residue class? My number theory book has hopelessly lost me on the topic of residue classes. I understand the very basics of congruence and modular arithmetic, but if someone could give not only a formal, but intuitive and helpful explanation. Also I need to understand why they matter, or I will likely have difficulty grasping the concept.
 A: Basically residue classes are the possible remainders. For example:
$a \equiv b \ (\mathrm{mod}\ m)$
We say, $b= [a]_m$
Now for the APPLICATIONS:
$\implies$ This is used in most of the famous theorems, for example: Fermat's little theorem
$\implies$ Modular arithmetic is often used to calculate checksums that are used within identifiers - International Bank Account Numbers (IBANs) for example make use of modulo $97$ arithmetic to trap user input errors in bank account numbers.
$\implies$Arithmetic $\mod 7$ is especially important in determining the day of the week in the Gregorian calendar. In particular, Zeller's congruence and the doomsday algorithm make heavy use of $\mod-7$ arithmetic.
And yeah, as you said it is used in clocks, too.
This is just a gist of the applications. Residue classes find applications in security systems. You will find more here .:)
A: I am unable to give an explanation that won't be considered straight up plagiarism, but if you can find Anderson and Feil's book "A First Course in Abstract Algebra" (which for a while I renamed "A First Curse in Abstract Algebra" till I started understanding bits and pieces of it) then I recommend their short chapter on Modular Arithmetic (chapter 3 in the second edition). 
I found their explanation - and even their proofs - of residue classes reasonably easy to follow, and it opens up the world of arithmetic on residue classes right away.
Interestingly a preview of that book up till page 33 can be found here and that just about includes their basic intro to residue classes. The missing part of the chapter is about arithmetic on residue classes - but if you get the first bit then the arithmetic can be understood from wikipedia or other online sources.
