TITLE: Difference between modulus and remainder
QUESTION [10 upvotes]: (I've come here from this question.)
What's the difference between modulus and remainder?
(Bearing in mind until 5 minutes ago I thought they were the same :P )

REPLY [5 votes]: Let's say we have integers $a $ and $b$.  The $remainder$ of $a$ divided by $b$ is the number $r$ such that $a = qb +r$ for some $q\in\mathbb{Z}$ (here $\mathbb{Z}$ is the set of integers), and $0\leq r <b$.  In this situation, $r$ is uniquely determined.  That is, there is only one value of $r$ which satisfies these requirements, and so we say that $r$ is $the$ remainder of $a$ divided by $b$.
Modulus, in this situation, refers to the number $b$.  However, given the context, I think what you meant to refer to was "$a$ mod $b$", or the class of $a$ modulo $b$, which is actually an infinite set of integers.  Precisely, the class of $a$ modulo $b$ is the set 
\begin{equation*}
a+b\mathbb{Z} = \left\{a+bn\;|\;n\in\mathbb{Z}\right\}.\end{equation*}
Every number in this set yields the same remainder after division by $b$.  So, for example, $5$ mod $7$ is the same as $12$ mod $7$, because we have the equality of sets
\begin{equation*}
5+7\mathbb{Z} = 12 + 7\mathbb{Z}.
\end{equation*}
Often, people say "$12 $ mod $7 = 5$," which is technically incorrect.  What we should say is "the class of 12 mod 7 is equal to the class of 5 mod 7", or "5 and 12 are equivalent modulo 7".  When working modulo $n$, we have a preferred set of representatives for each class modulo $n$, which is usually
\begin{equation*}
\{0,1,...n-1\}.\end{equation*}That means that every number is equivalent to one of these numbers modulo $n$.  When we use this particular set of class representatives, they just so happen to coincide with the usual notion of remainder.