Given a group $G$. Let $\mathbb{C}$ be the complex field. Then $\mathbb{C}G$ is the set of linear combinations of elements of $G$. Addition and multiplication are defined as usual. We can also think $\mathbb{C}G$ as the set of all functions from $G$ to $\mathbb{C}$. Addition is the addition of functions and multiplication of two functions is their convolution.
Now if we are give an algebraic group $G$ and $N$ is a subgroup of $G$. I saw the notation in many places. For example, cluster algebras and representation theory. Why $\mathbb{C}[N]$ is called coordinate ring of $N$? What are the elements of $\mathbb{C}[N]$? What are the additions and multiplications in $\mathbb{C}[N]$. Thank you very much.