Which grammar is this?
Technically it is a linear grammar, where every production is of the form $A\to uBv$ where $A,B$ are variables, and $u,v$ are terminal strings. The linear languages are in between the regular and context-free languages. 

The power of these grammars is reduced because in every string (sentential from) derived there is (at most) one variable. They are more powerful than right-linear grammars because the terminal strings that are generated during the derivation can appear both left end right from the variable. An example of a linear language that is not regular is $\{a^nb^n \mid n\ge 1\}$.
An example of a language that is context-free, but not linear, is $\{ w\in\{a,b\}^* \mid w \text{ contains the same number of $a$ and $b$}\}$.