Do you just plug in a NAS in to your router.
Does it have to be compatible with your make of route.
The short answer is no. It just talks 'ethernet' just like all devices in your network. As long as they comply to the standards. The longer answer is that in most cases it 'Just Works'™. But in rare cases there is an incompatibility between the switch and the NAS. (Just like sometimes a computer can't connect to a certain switch)
Are there any conflicts with operating systems?
Sometimes. The NAS is actually a small Linux server, and it runs one or more daemons to talk to the world. (Smb, ftp, http, afp, ...). Http and ftp are pretty low level, and never give problems. Although depending on the 'fancyness' of the webinterface it can give problems with certain browsers. But afp and smb are proprietary protocols, which have been reverse-engineered to be able to support it on Linux. It is stable, but the proprietor (Apple resp. Microsoft) can change it at any time. In the last few years a lot of old NASses were no longer accessible from new Windows systems, because MS dropped some authentication scheme in W7-8. And AFAIK Apple changes more often.
When the vendor of the NAS doesn't update the firmware, it's in that case no longer usable for that task. (Unless you hack the box and install your own OS).