Author markup means using the author property on Article, BlogPosting and other CreativeWork types to name the creator. Google's Article documentation describes best practices for it: use a Person or Organization type, put only the name in name, and use url or sameAs to point to a page that identifies the author. It replaced the long-retired rel=author and Google+ authorship system.
A short history
From 2011 to 2014 Google ran Google Authorship, which used rel=author links to Google+ profiles and showed author photos in results. Google ended it in 2014. Today there is no author photo rich result. Author markup helps Google and other parsers understand bylines, and it supports the broader aim of showing who is behind content.
Reviewers and contributors
Schema.org also offers reviewedBy on WebPage and contributor and editor on CreativeWork. These describe expert review accurately. Google has not documented any feature that uses them, so they matter for clarity rather than appearance.
Example
Multi-author posts
A compliance SaaS, Orrisvale, publishes a guide written by its security lead and a guest attorney. The markup lists both as Person objects in the author array, each with a url: one to the internal bio page, one to the attorney's firm profile. The visible byline shows both names.
Common mistakes
Stuffing titles into the name
"Dr. Jane Ruiz, VP Security at Orrisvale" in author.name makes the name harder to match. Keep it to the name.
Crediting a person in markup but not on the page
The byline users see should match the markup.
Questions
Google ended Authorship in 2014 and does not use rel=author for search features. Use the author property in structured data instead.