"Thrombocytopenia" is the medical term for low platelets, with mild thrombocytopenia they might not have any symptoms at all.
With lower amounts their blood wouldn't clot as well as it should, they might feel tired, have tiny red dots show up on their skin, bruise easily, have bloody gums or blood in their stool. Bleeding could worsen as the platelet count goes down.
A platelet count below 20,000 is considered a medical emergency.
http://www.nlm.nih.gov/MEDLINEPLUS/ency/article/000586.htm
