Why are Elephant Tusks not symetric?

An Elephant's tusks are long incisors - meaning it is a very large tooth - made of ivory. It is common for one tusk to be shorter than the other, which comes from the elephant using one tusk more than the other. Just like humans being either right or left-handed - elephants have a dominant tusk, often called the "master tusk."

What do Elephants do with their Tusks?

Tusks are the Swiss Army knife of elephants. They are used for digging for water, stripping bark from trees, lifting objects, and for defending. They are used to impress the females and intimidate other males. The tusks also protect the trunk. The tusk that an elephant uses more often will show the wear and tear. 

How do Tusks grow?

Tusks continue to grow along with the elephant aging, but once broken, damaged, or removed, they stay that way.

Interestingly - in African elephants both males and females have tusks, while in Asian elephants only the males do.