Given a monoid $M$, a ring $R$ and the monoid ring $R[M]$, which consits of all functions $f:M\rightarrow R$ with $f(m)\neq0$ for only finitely many $m\in M$. The (abelian) group structure is given by multiplication of functions and the convolution product $\ast$ as multiplication: With the characteristic function $\chi_m$ of $m\in M$ we can write elements of $R[M]$ as $$\sum_{m\in M}r_m\chi_m$$ for some $r_m\in R$ (mostly zero). Then we have $$f\ast g=\sum_{a,b\in M}r_a s_b\chi_{ab}.$$
This is what we have defined in the lecture. We see from this form that $f\ast g$ again is in $R[M]$.
Wikipedia says that the convolution product is $$(f\ast g) (m)=\sum_{a,b\in M,ab=m}f(a)g(b).$$
I guess this definitions are the same, but why?
Evaluate the $f\ast g$ defined by the first formula on $m\in M$. Every term $\chi_{ab}(m) $ will be $0$ if $ab\neq m$, so that the only terms that matter in the sum are those such that $ab=m$, in which case $\chi_{ab}=1$.
So you are left with $$(f\ast g)(m)=\sum_{a,b\in M}r_a s_b\chi_{ab}(m)=\sum_{ab=m}r_a s_b,$$which gives the result, as $$f(a)=\sum_{m\in M}r_m\chi_m(a)=r_a.$$