Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, and $\mathfrak{b}$ a Borel subalgebra. We work over a field $F$. Given a weight $\lambda\in\mathfrak{h}^{*}$, let $F_{{\lambda}}$ be the 1-d dimensional $\mathfrak{b}$-module on which $\mathfrak{h}$ acts by multiplication by $\lambda$, and the positive root spaces act trivially. Now, the Verma module $M_{\lambda}$ of the weight $\lambda$ is the $\mathfrak{g}$-module induced from $F_{\lambda}$, i.e.

Using the Poincaré-Birkhoff-Witt theorem we see that as a vector space $M_{\lambda}$ is isomorphic to $\mathcal{U}(\overline{\mathfrak{n}})$, where $\overline{\mathfrak{n}}$ is the sum of the negative weight spaces (so $\mathfrak{g}=\mathfrak{b}\oplus\overline{\mathfrak{n}})$. In particular $M_{\lambda}$ is infinite dimensional.

We say a $\mathfrak{g}$-module $V$ is a highest weight module if it has a weight $\mu\in\mathfrak{h}^{*}$ and a non-zero vector $v\in V_{\mu}$ with $Xv=0$ for any $X$ in a positive root space and such that $V$ is generated as a $\mathfrak{g}$-module by $v$. The Verma module $M_{\lambda}$ is a highest weight module and we fix a generator $1\otimes 1$.

The most important property of Verma modules is that they are universal amongst highest weight modules, in the following sense. If $V$ is a highest weight module generated by $v$ which has weight $\lambda$ then there is a unique surjective homomorphism $M_{\lambda}\to V$ which sends $1\otimes 1$ to $v$. That is, all highest weight modules with highest weight $\lambda$ are quotients of $M_{\lambda}$. Also, $M_{\lambda}$ has a unique maximal submodule, so there is a unique irreducible representation with highest weight $\lambda$. If $\lambda$ is dominant and integral then this module is finite dimensional.