Image source: Wikipedia.com
Hay fever, which does not refer to being allergic to hay, but rather being allergic to the pollen of ragweed in most cases. Hay fever may progress to asthma in up to 50% of people. Symptoms include a runny nose, sneezing, and tearing of your eyes.
Proteins in dead skin cells of animals can be a source of allergies too. Black cats are more allergenic than white cats apparently. Do keep in mind that cats can also bring around pollen and other things along with it! It tends to be the case that the longer the hair or fur, the more there is of the spreading of the dead skin cells.