There are a number of different ways to go about playing your classic ROMs and each way has it's own benefits and detractors. As for which method will appeal to you or perhaps more accurately meet your needs is up for you to decide. Never the less here are some suggestions to help you get started.
If you have a Raspberry Pi RetroPie is probably one of the best known and supported operating system options for retro gaming and for good reason too. It is very easy to setup and most importantly it is very well documented. The RetroPie Wiki is very well laid out and features clear and concise categories, want information about about a specific emulator just look under the "Emulators" category on the left side of the screen. It also has categories for Controllers, Emulation Station (The user interface RetroPie uses), and even a category for adding additional software called "Ports." Another option for the Raspberry Pi I find very good is Lakka which has support for a wide variety of systems including desktops.
RetroPie does have instructions for setting itself up on either Ubuntu or Debian which are what you would need to use if you want the RetroPie experience on a desktop system. This process is a bit more involved but once again the RetroPie Wiki will have you covered. Depending on the age of your system it may only have a 32-bit processor (CPU) in which case you won't be able to use Ubuntu because it no longer supports 32-bit CPUs thankfully Debian still does. The download section for Debian has a lot of options but for the sake of simplicity I'll just point out that if you are looking for a version of Debian that works with 32-bit CPUs you'll want to download the "i386" version of Debian. I would also recommend choosing the network install version as it will be a much smaller .iso (an .iso is an image file most commonly used to package operating systems) to download. If your system does have a 64-bit processor and you'd still like to use Debian you would then need the "amd64" version of the .iso. Don't worry if you've downloaded the wrong .iso if the image is not compatible with your CPU it just won't boot, nothing will break.
When using an old desktop for a retro gaming setup it is worth noting that even a 12 year old desktop may have a 64-bit processor and still be very capable of performing everyday computing tasks like web browsing and word processing especially if the CPU has 2 or more cores. Debian and Ubuntu are good operating systems but they use what is referred to as a fixed release cycle. This means that every 6 months or 5 years if you opt for a LTS (Long Term Support) version you will need to preform a new install in order to update to newer versions of certain programs. The upside to this method in theory is a more stable system, the downside is newer versions of programs may have new features you wont have access to until the next release cycle. This is not a big deal if you simply want to use the old system as a retro gaming console but if you want to do a bit more with your old system there are other alternatives.
Solus and Void Linux are known as rolling release distros (Linux distributions are often referred to as distros for short) this means that you install them once and then just update the system to stay up to date, no need to reinstall the operating system every 6 months or 5 years just to get updated versions of new programs. It is worth noting that the RetroPie install script will not work in Solus or Void Linux however both distros include RetroArch (the program that makes using emulators convenient for retro gaming) in there repositories (repos or repositories are like app stores for Linux based operating systems). Solus is a very user friendly desktop focused Linux distribution that is excellent for use with RetroArch, the only reason you wouldn't be able to use it is if the CPU is 32-bit. That being said Void will have you covered here as it supports 32-bit and 64-bit CPUs it even has an image available for the Raspberry Pi. Both Solus and Void have .iso images that feature the Mate desktop which is an excellent user interface that is light on resources and easy to customize. The Solus developers also maintain the Budgie desktop which is also very nice, in fact I'd say if your system has 4gb of RAM or more then use the Budgie .iso from Solus. Void also offers a base .iso image similar to what Debian refers to as a net install that will allow you to install a very minimal base system (no desktop) that you can build into what ever you want, it will require more work on your end to setup but the Void Wiki is very helpful and at the end of the day you'll have a retro gaming system configured to meet your needs.
These are just some suggestions with regards to retro gaming setups and rigs and as you can see a large part of what you want to do depends on what you have available to work with and how you want to work with it. Personally I like rolling release distros and the 12 year old desktop I have setup for retro gaming is powerful enough to run Solus with the Budgie desktop. For emulation I'm able to play my NES, SNES, N64, GBA, PS1 and PSP ROMs comfortably. If I want to go online I can do that easily and I can keep everything up to date with one simple command. Will this be the case for all 12 year old desktops? Probably not but you never know until you try, sometimes that old system you have may surprise you.