Most people would see there are two main types of computer. Work stations and servers. In most homes you will find a workstation and in most businesses you will find several work stations and several servers.
A server is a central point of access for resources that are going to be shared. To this end they tend to be much higher specd. They will typically have a RAID disk drive setup that will be mirrored. There will also be redunancy built in including extra power supply.
As far as cases go, then there are free standing and rack mounted. Rack mounted ones are long flat(ish!) boxes that you put in a rack (big cupboard).
Have a look at someone like Dell (click here) and the servers they offer. To get a better idea of specs/design.
I doubt there is much demand (need?) for self build servers.
Appoliges if this isn't the information you were looking for!