What does elevated immunoglobulin a mean?

It depends. There are several types of immunoglobulin, igg, iga, IgM and ige. Having a "high" level of each of them could mean different things. It is also important to know if it is a "high" overall level, or a "high" level of a specific kind of igg, iga, igm, or ige. In general, a high ige means "allergy". High igg/a/m can mean different things depending on their specificity (total or specific).