What kind of elements are in asteroids?

It depends on the type of Asteroid. You usually have 3 types of asteroids: Chondrite Asteroids, Stony Asteroids, and Metallic asteroids. 

Chondrites are usually the most common. They are the most ancient of asteroids found in our solar system and are usually composed of clay and silicate rocks.

Stony asteroids are usually made of silicate materials such as quartz, feldspar, mica, amphibole, pyroxene, olivine, etc. I think it's usually quartz that you find on asteroids, but those other minerals are also there.

Metallic asteroids are made of, well, metals. Usually, they are made of iron, but they are often a mixture of iron and nickel (80% to 20% respectively). But in some cases, they are also made of rarer metals than iron such as iridium, palladium, platinum, gold, magnesium and other precious metals such as osmium, ruthenium and rhodium.