Intel and AMD

With recent mesa and ffmpeg, both cards are supported by VA-API through libva>=2.1. Install libva-intel-driver and libva-mesa-driver. Make sure that your system does not have the following packages,

1. libva-vdpau
2. libva-va-gl
3. libvdpau-va-gl

Intel

You need to use hwdec=vaapi and other options can be configured according to your GPU. To make it permanent, add it to ~/.config/mpv/mpv.conf.

Intel HD graphics need i965-va-driver installed to work properly (at least on debian stretch).

AMD

Since dedicated graphics are going to be used, some environment variables are specified to instruct the driver to load specific modules.

DRI_PRIME=1            # for laptops with switchable graphics
LIBVA_DRIVER_NAME=r600 # value depends on the driver, check /usr/lib/dri for possible options

In mpv, you specify hwdec=vaapi and gpu-context=x11egl for Xorg.