The Washing Machine was added as part of The Kitchen Update (See also Dishwasher, Cookie Jar and Bar Stool).
The Washing Machine is an appliance which allows you to repair your armour, eg Chestplate. To power the machine, you need to put Soapy Water or Super Soapy Water in the top right slot. Simply put your armour into the corresponding slots on the GUI and hit the start button. Depending on the fuel you use, after every X amount of second(s), it will add one durability back onto your armour.
You can use the RecipeAPI to register custom armour from other mods to work with the Washing Machine.
Crafting Recipe
The crafting recipe is 7 quartz blocks, 1 glass pane and 1 furnace