My understanding of projective covers and injective hulls for modules over a (finite-dimensional) associative $\mathbb{C}$-algebra $A$ is as follows.
$\bullet$ The projective cover of $M$ is an epimorphism $f : P \to M$ such that $\text{ker}(f)$ is a superfluous submodule of $P$, and $P$ is a projective module. (Equivalently, $P$ is the minimal projective module such that $M$ is a quotient of $P$.)
$\bullet$ The injective hull of $M$ is a monomorphism $g : M \to Q$ such that $\text{im}(g)$ is an essential submodule of $Q$, and $Q$ is an injective module. (Equivalently, $Q$ is the minimal injective module that contains $M$ as a submodule.)
I'd like to understand the purpose of projective covers and injective hulls--how are they used, and why are they important? I'm also interested to know if the projective cover for $M$ is more valuable than its injective hull, or not.