It stands for "network attached storage", it's basically a standalone disk drive that is accessible to all devices within the local network (or public internet, if the device is setup that way).
In home setups, it's often used as a way to store terabytes of digital media (movies, videos, locally hosted wikipedia)
Just a computer with a bunch of hard drives so you can store your media all in one place. Most of the time people expose this to their home network so they can access the files from all their devices while on the same wifi, but you can also expose it to the internet so you can access the files anywhere.
Adding on to what others have said, I have one set up that's also used as part of my backup strategy for the important stuff on all the other boxes around here.
In home setups, it's often used as a way to store terabytes of digital media (movies, videos, locally hosted wikipedia)