"Allergic shiners" is the common name for periorbital venous congestion -- that is, dark "bruises" under the eyes caused by allergies. Swelling tissues in the sinus cavities prevent the small veins under your eyes from draining, and the blood collects. Because these are not proper bruises, they are usually pink, red, or purple, and do not develop the yellow penumbra of a true bruise.
There is often confusion between allergic shiners and rings and bags under a person's eyes; while these can be caused by allergies, they are often caused by old age, genetics, and the simple fact that the lower eyelid is predisposed towards hyperpigmentation (the technical term for shadows under your eyes is periorbital hyperpigmentation). This is not generally a medical concern, while allergies are.
Allergic shiners are a concern only in that they indicate a significant allergic reaction, and are treated by treating the allergy. They may also result from chronic upper airway congestion from other causes, and you should check with your doctor if they do not go away in the absence of seasonal allergies.