Let $R$ be a commutative ring, $a\in R$ and $b\in R-\{0\}$ we say that $b$ divides $a$ (notation: $b\mid a$) if $\exists t\in R$ such that $a=bt$.
In my book I often see the symbol $\dfrac{a}{b}$ and it confuses since it looks like usual division. But I guess that $\dfrac{a}{b}$ is the same as $t$. However, I think that the notation $\dfrac{a}{b}$ is informal. Am I right?
The symbols $\frac ab$ and $a|b$ are completely different things.
The first one usually denotes the equivalence class of pairs equivalent to the pair $(a,b)$ via the relation $(a,b)\sim (c,d) \iff ad=cb$. This is the same thing as $ab^{-1}$ and also $a/b$. In a nutshell, we are talking about "the result of $a$ divided by $b$" here.
The notation $a|b$ denotes that $a$ and $b$ are related by divisibility, i.e. that $ar=b$ for some suitable $r$. This doesn't produce any "result" like the last one: it just is a statement about how $a$ and $b$ are related.