Did you google the tests? From what I have read, IgA is in a group of tests run for antibodies. It is done for allergy testing, and is a test for Celiac disease (sensitivity to wheat, gluten, etc.) So in a sense is an allergy tests. If you go to Labtestsonline.org, you will find a complete explanation.
Of course I googled the test. :-)
The IgA deficiency is a form of immunodeficiency. For celiac, they often look at IgA levels of reaction to gluten. In fact, because I have an IgA deficiency, they're having to do a new round of Celiac testing looking at my IgG levels to see if I had a false negative test previously.
It's not really an allergy test, although looking at various IgA antibody levels to specific items can help identify allergies.
But allergies are a form of over-active immune system stuff, while this is saying my IgA levels are lower than they should be in general (not looking at antibodies for a specific disease or substance).