Dragonflies and damselflies are insects (3 body parts, 6 jointed legs, adults have wings); and both undergo 'incomplete' metamorphosis (i.e., hatch from egg looking like miniature, wingless adults. Eat, grow, shed; final shed reveals wings (which, in my opinion, is an amazing metamorphosis and shouldn't be called 'incomplete.' If I were queen...)
|Flame Skimmer Dragonfly|
You can usually tell dragonflies and damselflies adults apart by looking at their wings when perched. Dragonflies hold their wings perpendicular to their bodies; damselflies hold theirs parallel. If you get the chance, you can also tell by comparing abdomen size (that's the 'tail' end). Dragonflies generally have a thicker abdomen.
|Desert Firetail Damselfly|
|American Rubyspot Damselfly|
Whether dragon or damsel, though, Arizona Dragonfly is a great site with fantastic photos of these flyers.