AOSP is a term you'll see used a lot while reading about Android on the internet. It's the open-source side of Android, and the code every device running all versions of Android is based on.
AOSP stands for Android Open Source Project. The AOSP was designed and written by folks who had a vision that the world needed an open-source platform that exists for developers to easily build mobile applications. It wasn't designed to beat any other platform in market share, or to fight for user freedom from tyrannical CEOs. It exists as a delivery mechanism for mobile apps — like Google's mobile apps, or any of the 1,000,000+ in the Google Play store. Luckily, Google realized that using open-source software would ensure that this operating system/mobile application content delivery system is available for all, for free. And by choosing the licensing they did, it's also attractive to device manufacturers who can use it as a base to build their own mobile OS.
In total, there are more 550,000 files in AOSP — which lives at source.android.com — and it all checks in at over 18GB in size when downloaded.
The premise plays out rather nicely. Google writes and maintains a tree of all the Android source code — that's AOSP. It's made available for everyone (you, me, manufacturers you've never heard of and not just big players like Samsung or HTC) to download, modify, and take ownership of. The big players then use their modified version of this source to build their own operating system. Some, like Amazon, radically changed everything without a care to use Google's official applications and keep their device in compliance with Android guidelines. Others, like Samsung, radically change everything yet still follow the Android Compatibility program so they could include Google's core application suite — including the Google Play store. Still others, like the folks at CyanogenMod, enhance the pure AOSP code with additions but don't change the overall look and feel. Again — that's how this open-source thing works. You can't have it without allowing folks to change it as they see fit, for better or worse.
Any of us can download and build the AOSP. We can even stay compliant and pass the compatibility suite and contact Google about including their applications. Yes, any of us could build our own device using the AOSP code in our garage or basement with Google's full blessing. That's the beauty of the AOSP, and we wouldn't want it any other way.