Based on your post, it seems that UT2003 wants its binary to actually reside in /usr/local/games/UT2003. However, that location is rarely (if ever) in someone's $PATH (an environment variable that tells the shell where to look for the programs you type it). The installer wants you to place a symbolic link (similar to a shortcut, in Windows) to a location that does exist in the $PATH.
This way, instead of having to type the full path to UT2003 to run it, you'll just type something like "ut2003" on the command.
Before you can make a sym link, you'll need to know the name of the ut2003 binary. I'm going to assume its "/usr/local/games/UT2003/ut2003", but please verify that this is in fact the correct location.
Probably the best location for the symlink is /usr/bin/, so that's where I'm going to put it. If the installer suggests a different place, use that instead.
So, the following command will make your link:
ln -s /usr/local/games/UT2003/ut2003 /usr/bin
You'll need to be root for that.
After linking, you should be able to launch ut2003 by typing
ut2003 (not as root).
Hope that was helpful