How do elephant breed?

They take part in sexual reproduction. Unlike plants who use asexual reproduction. They copulate just like all other mammals.

The same way as people, dogs, cats, horses, ect...
They mate pretty much in the same way as any mammal does. The female is mounted from behind and then all the action begins. However there are some differences like the gestation period of the female elephant is nearly 2 years. The pack is sometimes lead by the matriarch and wandering bulls will mate with season females. For the rest of the time the bull elephants are solitary creatures.

Elephants are mammals, so they reproduce via sexual reproduction. The physical relations are similar to other four-legged mammals including dogs and horses.