Some games will run fine with your current set up, others may require updated drivers to get the best performance, and a few may refuse to run without the driver updates. Determine what graphics hardware you are using and follow the appropriate section for AMD graphics or NVIDIA graphics. As for you weirdos getting by on "Intel Corporation 82G33/G31 Express Integrated Graphics Controller" or somesuch (like me), try to keep up.
You must be sure that the regular user that will run steam games is at least at groups "video" and "audio".
# usermod -a -G video,audio second_user
Debian has a security approach that all is forbidden unless sysadmin explicitly allows. This schema is managed by "task" groups. The very first regular user created during installation is included in groups "video", "audio", "cdrom", "scanner", and so on. The next users created are not included at any special group. This way, next created users will not be able to execute steam. It fails with weird messages regarding "X bad number out of range." You must have to include them explicitly with commands or users management GUIs like kuser.
There are packages available for Debian 8 "Jessie", Stretch and Sid.
Packages are not currently available for Debian 7 "Wheezy". If you want to run Steam on a Wheezy system, you would need to either build a jessie chroot (look at schroot for this) or upgrade to Jessie first.
64-bit systems (amd64)
Instructions are for Jessie and Sid systems only. Do not do this on Wheezy or you will break your system.
Add a "non-free" component to /etc/apt/sources.list, for example:
# Debian Jessie
deb http://httpredir.debian.org/debian/ jessie main contrib non-free
Enable Multi-Arch and update the list of available packages:
# dpkg --add-architecture i386
# apt update
Install the steam package:
# apt install steam
Install 32-bit OpenGL libraries:
Installing the 32-bit OpenGL libraries may uninstall the 64-bit drivers you may have previously installed. Installing the AMD graphics or NVIDIA graphics drivers for your system is recommended. The appropriate OpenGL library will be installed along with those drivers. If you installed these drivers from jessie-backports be sure to also install the 32 bit library from jessie-backports.
Nvidia graphics hardware:
# apt install libgl1-nvidia-glx:i386
ATI/AMD graphics hardware:
# apt install libgl1-fglrx-glx:i386
AMDGPU graphics hardware: (tested on stretch)
# apt install xserver-xorg-video-amdgpu:i386 xorg-video-abi-23:i386 xserver-xorg-core:i386
Adapted from: [Debian Wiki][1]