NAS - Generic term for a device that connects to your local network, and provides central 'network attached storage' for your media. Some NAS boxes can do far more than this though.
Server - A device on your LAN that 'serves' data to network clients. This data can be whatever you want in whatever format you need as long as you have clients compatible with the format.
The term NAS has been adopted by many OEM suppliers for the small, low power devices that you just install the disks into and power on. Custom software on the device takes care of the rest, and it is almost a 'set and forget' system once configured.
A 'server' can be anything you want it to be. Your HTPC could act as a server (as well as doing other duties), a NAS can be a server, or a huge homebuilt box with 30TB can be a server but also be a NAS. The terms are almost interchangable.