There are several different types of that kind of sails. They have different names depending of the construction, and the direction of the wind witch they will sailing.
Symmetric spinnaker,is used for saling with the direction of the wind. Have large amounts of camber, making them nearly hemispherical. Spinnaker is hard to handle with the changing wind, and the crew needs to be focused. Spinaker requires more lines, and additionally a spinnaker boom. Used mainly in races. Manufactured from lightweight fabric, usually nylon with a wide range of colors.
Asymetric spinnaker (genaker), is quite more popular sail, have wider range depending of camber and construction. Easier to use than the spinnaker, it does not require extra ropes. This type of sail can swim together with the basic sails, allowing the boat to develop a higher speed. Manufactured from lightweight fabric, usually nylon with a wide range of colors.
The code 0 is a tight reaching sail. Luff is as straight as possible, and the sail is flatter than other asymmetric spinnakers. Due to the flatness of the code 0, it is usually made with a high modulus luff line for supporting strength, and of a heavier, less stretchy fabric than normal for a spinnaker. Due to the tight luff and flat cut, the code 0 can be fitted for roller furling.