There are several ways to make an ocarina:

• Making, with the aid of molds, two halves of an ocarina, put them together and "paste" or join them with more clay. Drilling the holes, making the sound hole and then glaze them and fire them. The molds can be the two empty halves or it can be clay in the shape of the ocarina, split and hollowed and then joined. For a step by step instructions on making a clay ocarina by half molds: How to make an Ocarina Making of an Ocarina
• Some ocarinas are made with liquid clay poured in an already complete ocarina mold, so that it will solidify and it'll be ready for drilling, making the mouthpiece, glazing and firing. The hole that remains from the pouring method is usually corked. For more info: Old Glory
• Wood ocarinas are handcrafted. The most common are Sweet potatoes and In-line wooden ocarinas.[1]Hind Ocarina E-How - How to make a bamboo ocarina
• Plastic ocarinas also exist and they follow the same process of plastic molds that most toys follow.