what is the relationship between vector spaces and rings? Can you show me an example to show how vector and scalar multiplication works with rings would be really helpful.

The relationship between a ring and its modules is the analogue of the relationship of a field and its vector spaces.
For a field (or even skewfield) $F$, the Cartesian product $F\times F\times \dots\times F$ of finitely many copies of $F$ is a vector space in the ways you are probably familiar with.
There is no reason you can't do this for $R\times R\times\dots\times R$ for a general ring $R$: defining scalar multiplication is exactly the same. 
The deal is that for general rings $R$ you call this a module, and in general there are a lot more modules that don't look like finitely many copies of $R$ (or even infinitely many copies of $R$.) That is why vector spaces are nice: because their structure is completely understood in terms of direct sums of copies of the field/skewfield.
