Cloud identification can be a simple or complex process, depending on how specific you would like to be.  You can start off by identifying cloud on how they look.  Here are the three basic areas of classification: 

Cumulus – Cotton Ball Looking Clouds
Stratus – Thin and typically low clouds
Cirrus – Whispy and high clouds

We can then further classify clouds by how high they are. 

Typically, high clouds (above 20,000 feet) start with the word “cirro.”  As said before, we already have cirrus clouds, but we can add a few more as well to this level:

Cirrus – Whispy and high clouds
Cirrocumulus – High clouds that look a bit fluffy
Cirrostratus – High clouds that are flat, but thick enough to block the sun

Midlevel clouds (6,000 feet to 20,000 feet) start with the word “alto.”  Here are the two that fit in this classification: 

Altocumulus – Midlevel clouds that look fluffy like cotton balls
Altostratus – Midlevel clouds that are flat, but can block out the sun if thick enough. 

The lowest level of clouds (below 6,500 feet) don’t really get any extra words added: 

Cumulus – Fluffy clouds that look like cotton balls
Stratus – Thin, low clouds
Stratocumulus – Thin layers of fluffy clouds. 

Clouds that produce rain get the word “nimbo” added to the name.

Nimbocumulus – Low clouds that produce rain
Cumulonimbus – Tall clouds that produce rain and are usually associated with thunderstorms

Check out this chart of clouds from the National Weather