There are a lot of earring back styles that you probably recognize but might not know the exact name of. Even more complicated, in jewelry making these styles are broken down even further creating sub categories. In order to understand all these styles, I've made an easy to refer to chart.
Fishhook / French Hook Earrings
These earrings have slightly different looks depending on the design but the concept is the same. This style uses ear wire bent in a hook shape with no back and instead utilizes the weight of the earrings to anchor the "hook" in place when in your ear. Like all jewelry fishhook earrings can come in many metal forms, most commonly nickel, aluminum, brass, copper, sterling silver, fine silver, and gold.
Kidney Ear Wire
Kidney ear wires use a similar concept that the fishhook uses only with a latch. The latch adds security to the earring without adding additional weight which could cause the earring to sit differently when worn.
Clip on earrings are perfect for those who do not want to permanently pierce their ears but would like to wear earrings. There are sub-style to the clip on earring but the most common is hinged and it uses the hinge tension to hold onto the ear lobe. Clip on earrings are a great alternative however, if you tend to wear heavier clip on earrings continuously you may want to consider having your ears pierced because they can damage the soft tissue over time.
A more expensive option to the kidney ear wire, lever back earrings are typically sturdier. Lever backs snap tightly over the ear wire creating a secure closure and is popular for more fine jewelry earrings.
Hoop earring closures continue to evolve but the hinged and lever back hoop style is more typically used. The hinged hoop secures the ear wire by snapping it into a latch. Like the lever back style, hinged hoop closures are relatively secure as long as the latch itself hasn't been damaged.
Lever Back Hoop
Using the same concept, the lever back hoop has a component that snaps over the ear wire to secure it in place. I personally find the hinged hoop a more secure closure because lever backs can loose tension over time and the hinge will no longer snap over the ear wire tightly.
Post earrings are typically thought of as a stud earring but posts are simply ear wires that use ear nuts to secure the earring in place. There are many styles of ear nuts ranging from metal to plastic to rubber. These are all push back styles. Posts can also have screw backs or lock backs as well.
There are certainly more earring back styles than I have covered but I've found these to be the most common. Many of the newer styles are variations of these anyway. I find that everyone tends to have their preference of earring backs whether its due to comfort, style, or materials.