Coriander is a herb, commonly called Cilantro in the USA. As a very rare allergy it is not well studied and there has been no confirmation of which allergens are causing the reactions.
Allergic reactions are most likely to be caused by Bet v 1-like proteins, which cause reactions in people sensitised to birch tree pollen and give oral allergy type symptoms.
Another suggestion is that plants in the Apiaceae family contain profilin proteins which can cause allergic reactions in uncommon circumstances.
I have put together some more comprehensive resources at http://www.allergyresources.co.uk/Coriander.php
The allergy resources page for coriander covers the key allergens, which allergic syndromes are linked to this allergy and other foods which are commonly cross reactive with coriander.
There is also regularly updated links to useful websites specifically for coriander allergies and the latest research papers on the topic.
If you have multiple allergies and are interested in how they are linked or want to find out more about food allergies outside the usual top 14 then you can follow me on Facebook or Twitter, where I share a food allergy card every day of the week.