modulo is the operation of computing the remainder obtained when one number is divided by another. modulo is often abbreviated as mod.

$𝐚$ mod $𝐪$ (i.e., $𝐚\text{ 𝐦𝐨𝐝𝐮𝐥𝐨 }𝐪$) is the remainder after dividing $a$ by $q$, which is always an element of $\{0,1,2,3,\cdots ,q-1\}$. For example, $7mod5=2$, because the remainder of dividing 7 by 5 is 2.

modulus: Given $𝐚$ mod $𝐪$, we call the divisor $q$ the modulus, whereas modulo refers to the operation.

Modulo Congruence ($\equiv$): $a$ is congruent to $b$ modulo $q$ (i.e., $a\equiv b\mathbf{\text{ 𝐦𝐨𝐝 }}q$) if they have the same remainder when divided by $q$. For example, $5\equiv 12mod7$, because $5mod7=5$ and $12mod7=5$. In mathematics, the notation $a\equiv bmodq$ is identical to $a=b(\mathit{𝑚𝑜𝑑}q)$, meaning that the remainder of $a$ divided by $q$ is the same as the remainder of $b$ divided by $q$. Note that this notation differs from $a=bmodq$, which states that $a$ equals the remainder of $b$ divided by $q$.

Congruence vs. Equality:

$a\equiv bmodq⟺a=b+k\cdot q$ (for some integer $k$)

This means that $a$ and $b$ are congruent modulo $q$ if and only if $a$ and $b$ differ by some multiple of $q$. For example, $5\equiv 12mod7⟺5=12+(-1)\cdot 7$