If you completely destroy your shield, you lose it forever and will have to buy a new one. If your shield is close to breaking, make sure you repair it back at the Bazaar or use a Revitalizing Potion. Otherwise don't use it if you wish to save it.
You can always upgrade your shields so they would be more durable and can take more damage (2 additional levels of upgrades) but they can still be destroyed.
One of the shields can auto-repair over time, the Sacred Shield (the third available at the Bazaar). It can still be damaged and destroyed like the others, you'll just have to take a break from using it to allow it to repair itself.
There is an indestructible shield too that you can get later in the game, the Hylian Shield. You can get that winning the Boss Rush mode.