What is the difference between Dragonflies and Damselflies? 

Dragonflies and damselflies belong to the insect order known as Odonata, meaning "toothed jaw" - their mouthparts are serrated. They are often referred to collectively as "dragonflies", however there are two different sub-orders.
Dragonflies are insects in the sub-order Anisoptera (meaning "unequal-winged"). Hind wings are usually shorter and broader than Fore wings. They are usually larger, strongly flying insects that can often be found flying well away from water. When at rest, they hold their wings out from the body, often at right angles to it. The eyes are very large and usually touch, at least at a point.

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Four-spotted Chaser Libellula quadrimaculata
Ruddy Darter Sympetrum sanguineum
Common Darter Sympetrum striolatum
Black-tailed Skimmer Orthetrum cancellatum
Scarce Chaser Libellula fulva
Keeled Skimmer Orthetrum coerulescens
Southern Hawker Aeshna cyanea
Norfolk Hawker Aeshna isosceles		The isosceles triangle (yellow) that gives the insect its latin name.

Damselflies are insects in the sub-order Zygoptera (meaning "yoke-winged"). All four wings are near enough equal in size and shape. They are usually small, weakly flying insects that stay close to the water margins or water surface. When at rest, most species hold their wings along the length of their abdomen. The eyes are always separated, never touching.

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Blue-tailed Damselfly Ischnura elegans	m	m	f
Common Blue Damselfly Enallagma cyathigerum
Banded Demoiselle Calopteryx splendens
Large Red Damselfly Pyrrhosoma nymphula
Red-eyed Damselfly Erythromma najas
Small Red Damselfly Erythromma viridulum
Small Red-eyed Damselfly Erythromma vridulum
White-legged Damselfly Platycnemis pennipes

Please note, I am not an expert on dragonflies. I just enjoy photographing them. If any ID or other mistake is spotted please let me know via the contacts page.