Soda ash has been very popular in the do-it-yourself community. It is a household chemical that is considered to be very safe, and is chemically closely related to its cousin baking soda, though harder to find and slightly more expensive. It is used in laundry detergents, soaping, making glass, water softening, pool care, and for adding dye to clothing.
Sodium carbonate (Na2CO3) is commercially made from common salt and limestone. It's pH is a high 11.6, compared to the more mild sodium bicarbonate (baking soda) of 8.4. The high pH and cheap production is what makes it valuable to water softening and pool care. It's unique chemical properties make it useful for laundry - "It competes with the magnesium and calcium ions in hard water and prevents them from bonding with the detergent being used" (Wikipedia). I have a nice recipe for the home-made laundry detergent here for your perusal.
When used for home dye projects, it can help to mordant fiber-reactive dyes to vegetable fabrics. That
translates to binds certain synthetic dyes to fabric like cotton, hemp, or bamboo. It's usually used as a pre-mordant.
When used in baking, it can replace baking powder/baking soda in order to leaven breads.
When used to make glass, it is added to the mixture of salts and silica before the melt process is started. It hardens and makes the glass less permeable to water.
For home swimming pools, it can be added to increase the water's pH which lessens the corrosive effects of chlorine. Indeed, finding sodium carbonate in the pool supply section of the local multi-mart can be the most cost effective way to purchase washing soda, with five pounds selling for under $15.
Historically, soda ash was made from leeching the ashes of certain seaweeds in a similar manner to potassium hydroxide production from wood ashes (read our post about it here). Before you ask, the common historically used plants were glassworts, saltworts, barilla, and seaweeds of the Fucus family (Wikipedia). Very little research has been done to investigate plant sources in the New World (maybe a survival project for another day?). It is mostly mined nowadays, but can be lab created in a similar manner as sodium hydroxide.