Will the games for Linux (like Quake 4) work in Ubuntu (the .run extension)?

In short, yes.

Ubuntu is based on Linux, so any game that has a "native" linux version (or "port") will run under Ubuntu as it will under any other distribution of Linux.

Wine won't destabilse your system. It's really just another application like any other Linux application.

Thanks. I'll try. Actually I downloaded this file "et-linux-2.60.x86.run" yesterday. It was Wolfenstain Enemy Territory, I think. A 280 MB file just to find out that it didn't install on my system.

ET does run on Ubuntu - I play it pretty much every week :)

Here is a guide to installing on Ubuntu:-

https://help.ubuntu.com/community/EnemyTerritory

Wow, the guide's cool. Unfortunately, I am at office now running a Windows XP. I'll go back home and try it. I admit, I didn't try the command line, as I am completely new to Ubuntu or any Linux, and am unfamiliar with the Ubuntu commands.
I will try and get back to you, so I am keeping this thread alive for now.
Thanks a mill for the help, of course.
Have a nice day.

Iam getting this message as soon as I type the command:

"fida@fida-desktop:~$ sudo sh ./et-linux-2.60.x86.run
Password:
sh: Can't open ./et-linux-2.60.x86.run"

I don't get it. The file was okay,The file size is 258 MB. I downloaded the full installer. But it isn't working. What to do?

Another time I got this message:
"Verifying archive integrity...Error in MD5 checksums: 505ed7cf3757b8d31343c5723e57b5f9 is different from b8b59bc515d86cc845fb52f5d2c1442".
Does this mean that the file was incompletely downloaded? Do I have to re-download it?

First one looks like you mistyped the file name or were in a wrong directory, try typing just a couple of letters and press 'tab'.

Second one, yes, the file may be inconsistent, you should redownload it probably.

For Ubuntu to follow the command stated here, ("sudo sh ./et-linux-2.60.x86.run"), the file has to be on desktop, right?; I mean as long as the Terminal says: "fida@fida-desktop:~$ ", right?
I am really sorry, to be asking naive questions; I AM new to Linux and Ubuntu, so I am having these problems. I know it's child's play to you.
Regards

The desktop corresponds to the ~/Desktop directory, not the ~ one ("home" directory).

So if the terminal says:
    fida@fida-desktop:~$
then:
    cd Desktop
    sudo sh ./et-linux-2.60.x86.run

If it still complains, make sure it is set to "Executable" or "Run" (right-click the file on the desktop and check the permissions tab).

Then if it still complains, it might be a corrupted download like you say.