How ARE fabric designs actually applied to fabric?  

In
Several ways.
Some are applied colour by colour, usually by screen (i'e screen printed)
Others as you say, by roller..
Modern digital technology allows several colours to be printed at once
Others are woven, these are usually known as jacquards

Most printed fabrics...
are screen printed. Small series are printed with flat screens that are moved at a given and equal distance over the unrolled fabric. The next colour is applied by changing the screen...and so on.
Industrially produced fabrics are printed with rotary screens, the principle is the same but the screen is a cylinder and rolls continiously against the unrolling fabric. One printing machine holds as many colour screens as there are colours in the print.
Flat screen printing always requires acurate regsitration (the fitting of one colour with the other. Therefore many mid century fabrics like Marimekko have either a white background and sufficient distance between the colours, or the pattern has an overlap that gives tolerance in the registration. Even industrially produced prints are often designed to allow for small differences in the registration due to tention in the fabric etc.

Sorry Robert...
We must have been answering this one simultaniously. You are taking up more recent technologies. They should be mentioned indeed. There are digital inkjet printers that reach fairly fast production.

Digital
Technology is very exciting.
By the time I go back to University next month (I lecture) we will have a full digital printer set up, so I can simply scan any design in, and get it printed onto a fabric ground of my choice

Thank you both...
Amazing what can be done with and without digital technology.