The 10 Major Cloud Types
| High | Cirrus, Ci. Cirrocumulus, Cc. Cirrostratus, Cs |
|---|---|
| Middle | Altocumulus, Ac. Altostratus, As. Nimbostratus, Ns |
| Low | Stratocumulus, Sc. Stratus, St. Cumulus, Cu. Cumulonimbus, Cb |

From: http://www.ucar.edu/educ_outreach/webweather/cloud3.html
This method of cloud classification was proposed by Luke Howard (1803) who named the clouds based on their form:
Cirrus - curl
Stratus - layer
Cumulus - heap
Nimbus - rain
and on their height:
High
Middle
Low
Contrails fall into the category of cirrus as they are high level clouds that consist entirely of ice crystals.
For more useful cloud descriptions visit:
The following book is also some recommended reading for cloud form and classification:
Earth from Dorling Kindersley. Pages 466 to473.
This reference contains good descriptions of cloud type and form and also contains some useful images for cloud classifications.