Compiling things in linux is pretty easy:
1) Ensure you have the correct external files for the libraries you need. If you want to compile MAME to run under X-Windows, you'll need the SDL libraries. Under Mandrake, use URPMI (Or "Mandrake Update") to search for the SDL and SDL-dev packages, and install them. Ditto for ALSA and ALSA-dev. If you don't have the development and compiler packages like gcc, g++, make, and so on installed, you'll need them too.
From here you'll need a terminal ("command prompt").
2) Change to the folder that has the advancemame source, and decompress it. "tar xvzf sourcename.tar.gz" will do that for you, and create a folder underneath the current one with the contents of the file.
3) Change into the newly created folder. At the command prompt, type "./configure" (without the quotes, of course). It's important to remember the "./" (dot-slash) bit at the front.
The autoconfiguration program will now run, figuring out where all your development programs and files live.
4) Once that's done, and if you don't get any errors, you're ready to compile! Simply type "make" and the program will compile (this will take quite some time). Once it's complete (and again, you don't see any errors) type "sudo make install".
To install programs under Linux, you must be an administrator (hence why it's difficult for viruses and spyware to work under linux, but I'll get off that soapbox for a minute). "sudo" is the linux command to temporarily promote your privaledges to a single command, and then revert your status. (It stands for Super User Do). It will ask you for a password, which will be the root password. Alternatively you can manually promote yourself to root using the "su" command, run "make install", and then exit out of root's login after that.
Now, assuming you've had all the right development files installed and get no errors, advancemame will now be compiled and installed.
If you get any errors along the way, post them here and I'll help you track down the cause, and get you up and running.