The Computer Case is the most basic computer you can build, and usually what you'll get started with. The different tiers of computers have a different layout of slots available for parts, as well as increasing level of supported component tier (indicated by the roman numeral in applicable slots).
To get a running computer you‘ll need to install a CPU and some memory (RAM) in a computer case. You’ll usually also want to install a graphics card to be able to display text on a connected screen, and a hard disk drive (HDD) to install an operating system on (after booting from an OS floppy from an external disk drive for the first time, for example).
Computers can be upgraded using several different cards. For example, using a redstone card computers are able to read and emit redstone signals. Using network cards they can send and receive network messages, allowing them to communicate with other computers.
If a computer stops working / crashes, try using the analyzer tool on it (sneak-activate) to see the error message that caused the computer to stop.
Computer cases can be dyed using any dye. This is purely for aesthetic reasons, and has no practical implications.
Computer cases also serve as the base part when building robots in the assembler. The different tiers determine the available slots for building the robot. When using a creative case, the robot will also provide itself with power, as does the normal creative case.