Let $G$ be a group and $N$ normal in $G$. I have read about a map $\alpha : G\rightarrow \frac{G}{N}$ called the reduction map mod $N$. I would love if someone could please explain this to me.
Is it just the map that sends all $g$ to $gN\in \frac{G}{N}$ ?
It is the canonical group homorphism $G \to G/N$, $g \mapsto gN$ which actually belongs to the definition of the quotient group $G/N$. It is usually called the projection or quotient map.
Often the term "reduction map" is used for maps which are derived from the quotient map. For instance, there is a reduction map $\mathrm{GL}_n(\mathbb{Z}) \to \mathrm{GL}_n(\mathbb{Z}/p\mathbb{Z})$ which is induced by the ring quotient map $\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$.