 Solution

Expand $L$ to $L(M)$ by a constant $c_m$ for each $m\in M$. The diagram of a structure $\mathrm{Diag}(M)$ contains all atomic and negated atomic $L(M)$-sentences true in $M$. The elementary diagram $\mathrm{ElDiag}(M)$ contains every $L(M)$-sentence true in $M$.

The method of diagrams combines one of these sets with another theory and applies the compactness theorem. A model of $\mathrm{Diag}(M)$ yields an embedding of $M$ by $m\mapsto c_m$, while a model of $\mathrm{ElDiag}(M)$ yields an elementary embedding.