There are many ways of transferring digital images on to fabric so that the end results are washable. I'll give you the methods of a couple now and will add more later.
Basically there are two types of printers in common use at home, laser printers and inkjet printers. It is possible to print directly on to fabric using either type but first your fabric needs to be attached to a backing sheet of some sort to stiffen it enough to feed through the printer's feed mechanism.
There are two easy ways to do this: you can iron your fabric on to freezer paper cut to the size your printer uses or you can use A4 labels stuck on to your fabric. In both cases the fabric needs to be trimmed neatly to the size of the paper backing.
Notes for Laser Printers:
Fabric for these printers doesn't require any pre-treatment. It does, however, need heat setting after printing. Cover the print with some clean newsprint or other cheap paper and iron with a hot iron. Move the paper to a clean section and repeat. Usually two or three repeats are sufficient to remove the excess toner. After this the fabric is washable.
Notes for Inkjet Printers:
You need to be aware what type of ink your inkjet printer uses - there are two basic types, dye based inks and pigment based inks. The general guidance is that dye based inks will almost certainly need pre-treatment with something like Bubble Jet Set 2000 before printing (see their web site for instructions www.cjenkinscompany.com), otherwise the ink will wash out easily. Pigment based inks may not need pre-treatment - I've had excellent results with Epson's Durabright ink. I've printed directly on to cotton with this ink and have put it through the washing machine on a 40° cycle with regular washing liquid without any noticeable change in colour.
THERE ARE MANY DIFFERENT BRANDS AND TYPES OF INK AROUND SO TEST BEFORE YOU EMBARK ON ANYTHING MAJOR. THE ABOVE ARE GUIDELINES ONLY!