Dragonflies belong in the order Odonata. In that group there are two main groups, the Anisoptera or dragonflies and the Zygoptera, or damselflies. There is actually a third small group called the Anisozygoptera and is represented by two species found in Asia.
We have a number of damselflies here in Iowa We have 43 species of damselflies (if I got my quick count right) and 66 species of dragonflies. There is a great reference, the Iowa Dragonflies and Damselflies web site.
Here is a spreadwing damselfly–I think it is the southern spreadwing, Lestes australis , although there are some characteristics that cannot be examined in this photo. It is quite a beautiful creature.