Washing soda (also known as sodium carbonate or soda ash) used to be a common household product when people made their own soap. It is closely related to Baking soda (sodium bicarbonate), and it is actually made by heating baking soda. Washing soda was first made from salt and limestone to create a cheaper more accessible way for people to bathe daily. Washing soda can be used for everyday household uses such as cleaning and added to homemade detergents.
The first obvious use for washing soda that I have mentioned before is laundry detergent. The borax, fels naptha, and washing soda combo has been a tried and true way to clean clothes for a long time.
Another way washing soda can be used is for household cleaning. Although I usually just grab a spray bottle of vinegar for cleaning up spills and messes, washing soda can be used for grimier tasks (think coffee stains, grease, blood, soil). I have also heard it is good for cleaning out greenhouses (all the slime and goo that tends to grow there).
I am not sure how effective it is but I have also heard that a washing soda paste (just soda and water mixed together) will strip paint off surfaces.
I would caution that washing soda is not child friendly. It can cause eye irritants and will make one sick if ingested so be careful using this product around small children.