Much like how the Hamiltonian $H$ determines the dynamics of a closed system via $\dot\rho=-i[H,\rho]$, the Lindbladian (or Liouvillian${}^1$) is an extension of Hamiltonian generators to model the interaction of a system with the environment. More precisely, every Lindbladian $L$ consists of the closed system generator $-i[H,\cdot]$ and adds another linear map $-\Gamma$ onto it. In other words---to connect this to system dynamics---the system evolves according to $\dot\rho=-i[H,\rho]-\Gamma(\rho)$ so the solution is no longer $\rho(t)=e^{-i[H,\cdot]t}\rho_0=e^{-iHt}\rho_0e^{iHt}$ but now reads $\rho(t)=e^{tL}(\rho_0)$ and has no nice closed form anymore. To summarize the basics: a Lindbladian is not a Hamiltonian, it extends the Hamiltonian (more precisely: the Liouville-von Neumann generator $-i[H,\cdot]$) to model dissipative effects.
Either way, to ensure that $e^{tL}$ maps quantum states to quantum states the map $\Gamma$ of course has to satisfy additional conditions. In their famous seminal paper Gorini et al. proved that $L$ gives rise to "valid" dynamics if and only if there exist square matrices $\{V_j\}_j$ such that
$$
\Gamma(\rho)=\sum_j\frac12(V_j^*V_j\rho+\rho V_j^*V_j)-V_j\rho V_j^*
$$
for all inputs $\rho$; to add to the confusion, the $V_j$ are sometimes called Lindblad operators (in contrast to the Lindbladian $L$ for which the $V_j$ are the "building blocks"). Equivalently, there has to exist a completely positive map $\Phi$ such that $\Gamma=\frac12\{\Phi^*({\bf1}),\cdot\}-\Phi$ (where $\Phi^*$ is the adjoint of $\Phi$).
Having recapped all of this, a Davies generator is a special type of a Lindbladian, i.e. a Davies generator $D$ is of the form $-i[H,\cdot]-\Gamma$ and it satisfies some additional conditions. These read as follows:
- $[H,\cdot]$ and $\Gamma$ have to commute, that is, $\Gamma(H\rho)-\Gamma(\rho H)=H\Gamma(\rho)-\Gamma(\rho)H$ for all $\rho$
- It satisfies the detailed balance condition, i.e. ${\rm tr}(\Gamma(\rho)e^{-H/T}\omega)={\rm tr}(\rho\Gamma(e^{-H/T}\omega))$ for all $\rho,\omega$. This is basically self-adjointness with respect to a weighted inner product which guarantees that $\Gamma$ has only real eigenvalues.
Depending on the context sometimes one imposes yet another condition which is ergodicity meaning the kernel of $L$ has to be one-dimensional and spanned by $e^{-H/T}$. Anyway the reasons Davies generators come up are that they come from physical considerations ("system weakly interacting with a thermal bath") and that, on top, the additional conditions make them a handy and easy-to-analyze subclass of Lindbladians.
${}^1$ Strictly speaking Liouvillian is the general term for the generator $\mathcal L$ of a master equation $\dot X=\mathcal L(X)$, but in the case of open quantum systems---due to the requirement of complete positivity---this boils down to the Lindbladian which is why these names are often used interchangeably in the quantum literature
