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I'm trying to install the nvidia-driver for Debian.

I've read everywhere that the correct solution is to run sudo apt install nvidia-driver and the driver should install itself without problems.

However this command leaves me with the output

Reading package lists... Done
Building dependency tree 
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 nvidia-driver : Depends: nvidia-driver-libs (= 375.82-1~deb9u1) but it is not going to be installed
 Depends: nvidia-driver-bin (= 375.82-1~deb9u1) but it is not going to be installed
 Depends: xserver-xorg-video-nvidia (= 375.82-1~deb9u1) but it is not going to be installed
 Depends: nvidia-vdpau-driver (= 375.82-1~deb9u1) but it is not going to be installed
 Depends: nvidia-alternative (= 375.82-1~deb9u1)
 Depends: nvidia-kernel-dkms (= 375.82-1~deb9u1) or
 nvidia-kernel-375.82
 Recommends: nvidia-settings (>= 375) but it is not going to be installed
 Recommends: nvidia-persistenced
E: Unable to correct problems, you have held broken packages.

I've tried installing the missing dependencies (like sudo apt install nvidia-driver-libs) but this just results in

Reading package lists... Done
Building dependency tree 
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 nvidia-driver-libs : Depends: libgl1-nvidia-glvnd-glx (= 375.82-1~deb9u1) but it is not going to be installed or
 libgl1-nvidia-glx (= 375.82-1~deb9u1) but it is not going to be installed
 Depends: nvidia-egl-icd (= 375.82-1~deb9u1) but it is not going to be installed or
 libegl1-nvidia (= 375.82-1~deb9u1) but it is not going to be installed
 Recommends: nvidia-driver-libs-i386
 Recommends: libopengl0-glvnd-nvidia but it is not going to be installed
 Recommends: libglx-nvidia0 (= 375.82-1~deb9u1) but it is not going to be installed
 Recommends: libgles-nvidia1 (= 375.82-1~deb9u1) but it is not going to be installed
 Recommends: libgles-nvidia2 (= 375.82-1~deb9u1) but it is not going to be installed
 Recommends: libnvidia-cfg1 (= 375.82-1~deb9u1) but it is not going to be installed
 Recommends: nvidia-vulkan-icd (= 375.82-1~deb9u1) but it is not going to be installed

How do I install the nvidia-driver with apt?

You need to enable the non-free repositories:

sudo sed -i.bak 's/stretch[^ ]* main$/& contrib non-free/g' /etc/apt/sources.list

Then run apt update and try your installation again. You’ll probably also need to install the kernel headers if you haven’t already:

sudo apt install linux-headers-$(uname -r)

See the full instructions on the Debian wiki.