An unfired bowl is made on a Potter's wheel. Creating an unfired bowl requires a minimum of level 8 Crafting and gives 18 experience. In order to make one, clay must be used on a vessel carrying water such as a jug of water or a bucket of water to be made into soft clay. Once the soft clay is made, you must use the soft clay with a potter's wheel and select the bowl. Among other uses, it is needed to complete Dragon Slayer.
For the quest Dragon Slayer, an unfired bowl is required as one of four items in order to obtain Thalzar's map piece from the Oracle on Ice Mountain. The Grand Exchange may be used to buy items (such as the unfired bowl) although it will almost always be faster for the player to buy soft clay and make the bowl at Barbarian Village.