What exactly are
.run files and what are they used for? I've seen a couple of those lately and I'm simply wondering what file format it is used for.
A .run file is normally a custom made program which needs to be executed in order to install a program. These are not supported generally as they don't track where files go and don't normally provide an uninstall method. there is no way to be sure what the script will do to your system so they're considered unsafe.
They are close to the windows exe file and as such come with the same issues. If you know what you're doing and are happy about taking the risks, you can execute them with these commands:
chmod 755 programinstall.run sudo ./programinstall.run
They are usually executables of some form. They're not neccessarily installers although most of the ones you'll see are.
As Martin says, and assuming we're talking about an installer, they're usually not as good as packages for when you want to remove them however there are some installers (I'm thinking mainly of the Loki and LGP game installers) do track where they install to and provide a pretty robust uninstall binary.
And what Martin says will work for most binaries, most .run files I've seen are hybrid shell script and compressed binary elements. This means you can just run them through
sh. An example with the NVIDIA driver installer:
sudo sh NVIDIA-Linux-x86_64-260.19.06.run
You can check to see what a file really is (well it works most of the time) with
NVIDIA-Linux-x86_64-260.19.06.run: POSIX shell script text executable