Momentum conservation: $$\rho\frac{Dv}{Dt}=\nabla\cdot\sigma+\rho g$$
Mass conservation: $$\frac{D\rho}{Dt}+\rho\nabla\cdot v=0$$
What does $\frac{Dv}{Dt}$ and $\frac{D\rho}{Dt}$ notion mean here and how this derivatives can be re-expressed in term of partial derivatives?
Suppose that you have a differentiable function $f:\mathbb R^n\times\mathbb R\to\mathbb R$ of $n+1$ variables. If $g:\mathbb R\to\mathbb R^n$ is a differentiable curve in $\mathbb R^n$, you can compose it with $f$ to obtain a differentiable function of a single variable $h:\mathbb R\to\mathbb R$, namely
$$h(t) = f(g(t),t)$$
The derivative of $h$ can be expressed in terms of the differential of both $f$ and $g$ by using the chain rule, viz.
$$\frac{\text dh}{\text dt} = \nabla_{\mathbf x}f\cdot\frac{\text dg}{\text dt} + \frac{\partial f}{\partial t}.$$
When $g$ represents the flow over time of physical points, e.g., the points of some material evolving in time, its derivative is a velocity field $\mathbf v$ and it is customary to use the notation of the OP to denote the variation of $f$ over time along the flow. This is known as the material derivative. Indeed, it is just a total derivative with respect to the "time" variable $t$.
$D$ stands for covariant derivative. The equation is the covariant form of the continuity equation.
The total derivative. Essentially if you have a multi-variable function dependent on position coordinates and also say a scalar like time, then it's total Derivative is the sum of the partials of the functions with the position coordinate plus the partials of the function with time (i.e: the scalar coordinates)
