Authority refers to the reputation and / or expertise of the source's author (often the source's publisher, too). Let's explore those ideas further:
What makes the author an expert on this topic? What kind of educational, work, or other experience do they have?
Tip #1: Authority isn't the same in every situation. Make sure the person's expertise matches what they're writing about. Ex. Consider what makes a neurologist an expert in neuroscience, and what makes a knitter an expert at knitting. Each person took different paths to get their expertise. A neuroscientist is not necessarily an expert knitter.
Tip #2: In academic publications (scholarly journals, books, etc.) you will often see their name associated with a college, university, company or research lab. In most cases, this is the place where that person works. Here is an example of three authors that work at four different universities:
What is the reputation of the publisher?
Tip #1: University presses (like Harvard University Press) are usually very reputable overall. They mostly publish scholarly work, but sometimes publish books aimed at a more general audience.
Tip #2: There are a number of other reputable publishers besides university presses. Some publishers focus mostly on academic publishing (ex. Springer) but some "popular" publishers (ex. Harper Collins) also publish well-researched non-fiction that is aimed toward a more general audience.
Tip #3: If you're not sure if a particular publisher is a "good" publisher, ask your professor or a librarian. As you gain expertise in your field, you'll learn who the good publishers are.
In order to figure out whether someone is is an authority, you might have to look at a person's profile and search Google to discover their education and expertise.