The author is the person who published the species name first. The key word here is published. The name is usually followed by the publishing date however for some reason myFOSSIL didn’t want to follow that convention. If the genus name was later changed but the species name was not, the author will be in parans. So here is an example of an author: (Agassiz, 1843). The author was Agassiz who published the species in 1843, but later the genus name was changed. In myFOSSIL that would be either Agassiz or (Agassiz).