Matt and Devin's guide to Desktop Operating Systems
Arch Linux is essentially "Build your own Linux." With Arch, you get a base Linux installation where you must install everything else you'll need. When you first boot the system, you have to tell it to look for drives. You have to manually partition the drives and update your file system table (fstab). If you want a piece of software, you put it there. If you don't, it won't be there. You do have the niceties of modern Linux distributions (e.g. a package manager) but your experience is what you make it. If you need help, their community forums and wiki are among the best in the Linux world. In fact, users on many other Linux forums link to the Arch wiki for information. As far as software is concerned, Arch stays close to upstream and uses a rolling release model to keep your system stable but up-to-date.
Gentoo is like Arch on hard mode. With Gentoo, you build everything from source. You control the compiler, so you control your system. Depending on your source for source code, you may experience issues with stability. This can sometimes be helped by changing how you build/compile the apps, but may require getting an older version of the source code. It's a bit tough to get set up if you don't know what you're doing, but their community has a lot of help available to get you up and running and help with any problems you may encounter.
Arch is for people who want to dive deep into Linux. You build your own system, so if you don't know what's going on under the hood - you'll learn. Gentoo takes that a step further. Not only do you build your own system, you compile all your apps from source. Needless to say, these distributions are for the experts and the curious. The people who know what they're doing and the people who want to learn how it all works will benefit the most from these systems.