Damselflies and dragonflies are both odonates, but they belong to separate suborders. In damselflies, the fore and hind wings are similar in shape and narrow to a stalk at the base, and most species can bring them together above the abdomen at rest. In dragonflies the hindwing is broader at the base than the forewing, and the wings stay spread.
Other differences follow the same split. Damselfly eyes are set well apart on either side of a wide head; dragonfly eyes are large and usually meet or nearly meet on top. Damselfly larvae breathe through three leaf-shaped caudal gills at the tail tip, while dragonfly larvae use gills inside the rectum and can jet water from them for a burst of speed.
The familiar rule has exceptions. Members of the family Lestidae, the spreadwings, hold their wings partly open at rest, so posture alone misidentifies them. Eye placement is the more reliable field character.