I would say something like "$a$ is equivalent [or congruent] to $b$ mod $n$"—"mod" here being pronounced to rhyme with "rod" or "pod" etc. Some people say "equal" instead of "equivalent"; as long as the "mod $n$" part is there, it doesn't much matter.

It just now occurred to me that you also asked what the term "modulus" or "modulo" means. Practically, if we say that

$$ a \equiv b \bmod n $$

we mean that $a$ and $b$ both leave the same remainder when divided by $n$; that is, there exist integer values $p$, $q$, and $r$ such that

$$ a = pn + r $$ $$ b = qn + r $$ $$ 0 \leq r < n $$

It's called "modular" arithmetic from Latin modulus "little measure"; the implication is that the modulus $n$ is what you use to measure out the quantities $a$ and $b$. What's left (if anything) is the same in either case.