Let $\cal C$ be a category and let $T \colon \cal C \longrightarrow C$ be a monad. An $T$-algebra $A$ is presented by generators $G \in \cal C$ and relations $R \in \cal C$ if it is the coequaliser of two morphisms $TR \longrightarrow TG$. See here for more information. Unfortunately, the site does not go into much detail.
I am a bit confused about how this works in practice though. For example, consider the group $$ A = \langle x,y,z \mid xy=yx \rangle $$ Let $G = \{x,y,z\}$ and let $R = \{ * \}$ be a singleton. Let $F \colon \mathsf{Set} \longrightarrow \mathsf{Set}$ be the free group monad. Define $s(*) = xy \in FG$ and $t(*) = yx \in FG$. Then $s,t$ extend to group homomorphisms $FR \longrightarrow FG$, and I think that $A$ is the coequaliser of these.
Is that the correct way to think of this?