Flare is unwanted streaks of light on your photos caused by lens imperfections and/or pointing the camera at the light source.
A major problem in lens design is that of minimizing reflections from the surfaces of the lens elements. Such reflections result in multiple images and in light from bright objects outside the field of view arriving at the film. The various optical effects that can occur due to internal lens reflections are known collectively as 'flare'.
Reflections occur at boundaries where there is a sharp change of refractive index (e.g. on going from air into glass) - the sharper the boundary the stronger the reflection. Modern lenses have anti-reflection coatings, which make the change of refractive index more gradual and so reduce (but do not eliminate) flare.
All compound lenses will exhibit some flare when shooting into the sun. Strong reflections from scene highlights will also cause visible flare. Some lenses can be fitted with an external 'anti-flare hood’ that is designed to cut off all light-rays which do not originate from objects within the field of view.