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.

Basically residue classes are the possible remainders. For example:

$a \equiv b (\mod m)$

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 finds applications in security systems. You will find more here .:)