$\alpha = (\alpha_1,\cdots,\alpha_n), \, |\alpha| = \alpha_1 + \cdots + \alpha_n,$, and this is how $D^\alpha$ is defined in the notes I’m using to study distribution theory:
$$D^{\alpha} = \left(\dfrac{\partial}{\partial x_1}\right)^{\alpha_1}\cdots\left(\dfrac{\partial}{\partial x_n}\right)^{\alpha_n} = \dfrac{\partial^{|\alpha|}}{\partial x_1^{\alpha_n }\cdots \partial x_1^{\alpha_n }}$$
Can someone tell me in a bit more depth what this notation stands for?
It's defined in the image you attached... it is a higher order partial derivative. For example if $n=2$ and $\alpha = (1,2)$ then $D^\alpha f = \frac{\partial^3}{\partial x_1 \partial x_2^2} f$.