The Grothendieck group on $X$ is defined (in your notation) as $K(\mathcal{B})$, where $\mathcal{B}$ is the monoid of f.g. locally free sheaves on $X$, (i.e. vector bundles on $X$.) The group $L(\mathcal{A})$ is usually denoted as $G(X)$, or $K'(X)$. If $X$ is smooth, the embedding $\mathcal{B} \subset \mathcal{A}$ induces an isomorphism from $K(\mathcal{B})$ to $L(\mathcal{A})$ (i.e. from $K_0(X)$ to $G(X)$.)

Chuck Weibel's notes for his K-theory book (which can be found on his website) is a good place find some of this information.

The Grothendieck group on $X$ is defined (in your notation) as $K(\mathcal{B})$, where $\mathcal{B}$ is the monoid of f.g. locally free sheaves on $X$, (i.e. vector bundles on $X$.) The group $L(\mathcal{A})$ is usually denoted as $G(X)$, or $K'(X)$. If $X$ is smooth, the embedding $\mathcal{B} \subset \mathcal{A}$ induces an isomorphism from $K(\mathcal{B})$ to $L(\mathcal{A})$ (i.e. from $K_0(X)$ to $G(X)$.)

Chuck Weibel's notes for his K-theory book (which can be found on his website) is a good place to find some of this information.