A “legal authority” is just a written text (whether in electronic or print form) that says something about the law. Don’t confuse it with “the authorities”—people or agencies with legal power of some kind—or with “authority”—legal power to do something. An authority is a source in the sense that the texts you consulted when writing undergraduate papers were sources, but don’t confuse a legal authority with a “source of law.”
Authorities consist of texts of two kinds: primary and secondary authorities. “Primary authority” just means that a text is binding as law, at least over some people. In other words, it creates legal obligations or consequences. This includes statutes that legislatures have adopted, court opinions adjudicating disputes, and administrative regulations that an executive authority has promulgated. “Secondary authority” consists of everything else, including commentaries, model statutes, restatements of the law, etc. There is also a distinction between mandatory and persuasive authority. “Mandatory authority” is primary authority that actually governs your problem, question, or client in this case; “persuasive authority” is everything else.
For example, in a Texas hit-and-run case, mandatory primary authority would probably be Texas statutes and court opinions. Persuasive primary authority might be court opinions from other states; those opinions are binding on folks in those other states, but Texas courts may or may not find them persuasive. In the same situation, all secondary authority is, at most, persuasive. So if a leading professor at Texas A&M University School of Law wrote what many consider the definitive article on Texas hit and run, it would be secondary authority, and by definition persuasive; no court anywhere would be bound to follow her reasoning.