An appliance server is generally a (largely preprogrammed) server that performs specific functions. The common home LAN router is a simple appliance server.
Home routers today usually aren’t just routers (i.e. network controllers that relay packets from one computer to another) but typically act as NAT devices (letting multiple computer share one connection , and usually acting as a bit of a firewall by blocking connections that didn’t original inside the LAN), DHCP servers (i.e. getting an IP and other system settings from your ISP and assigning IPs and setting to the computers on your LAN) and may have other added functions like file serving (if you add a hard drive), email filtering, antivirus/trojan features, etc.
But for all these features, they aren’t called up to do nearly as many things a more general purpose computer can. They are limited to the software that is flashed into them, and usually have quite limited CPU power, RAM, etc. On the plus side, since their task is so predefined, they can be as reliable (and in principle, secure) as any of the other computer controlled devices in your life – your TV, your car, your toaster, etc. You control it through a menu system (like your TV) and external status lights (like your car) – and in the worst case, the on/off switch, the power plug or a reset button (as in many modern gizmos).
I can see where a fileserver network appliance could be useful. It’s basically a gadget that lets all your computers share a common hard drive over the network, so different teams can work on the seame file or check progress, without shuffling multiple, possibly outdated, copies of the file around the network. Similarly, A printer server network appliance could let all your users share printers (or switch to an available one if their local one goes down). Email is also a simple function suitable for a network appliance. There are unified network appliances that will handle all these functions and more.
The alternative is to use a “full featured” server for all of these functions, with all the foibles that we know so well: a more cumbersome set-up/install; the need for a sysadmin who knows what he’s doing; perhaps a greater possibility of misconfigurations, crashes and “hacking”. A full-featured server can be much more capable than an appliance (e.g. a full featured back-up strategy and separate working and development environments), but appliances are often a better entry solution --pretty close to plug and play-- and are cheap.
However, I’m not certain that an “application server” (containing a copy of AutoCAD that will be used by multiple users) is considered a network appliance yet. I expect that they will be, but I’m not sure that they are yet. It’s just a matter of nomenclature and state of the art. There’d be a thin line between a top of the line "everything but the kitchen sink"appliance and a really well-configured full-featured server running on high reliability hardware (e.g. “real” servers often use RAM with more bit --say 36-bit memory chips instead of 32, with the extra 4 bits used to test for errors). My acid test for calling it an “appliance”? If you accidentally kick the plug out of the wall and plug it back in, will it pretty much pick up where it left off? If it hangs or crashes, and Joe from Graphic Arts yells hits reset, would it be a crisis that could get him fired, or just an inconvenience that would get him yelled at? (not that this is a policy I advise, nor should an appliance need rebooting more than a few times a year, tops)
I hope that clarifies things. It really sounds like you need to call in a consultant, if only to help you choose and configure the right option for your needs.