1. A reading that focuses on the quality of the reporting. The copy editor looks at the copy from the perspective of the newspapers' readers. Are there unanswered questions -- questions readers would have AFTER reading the story? Are there gaps in the story? The copy editor should make sure that the questions are answered and the gaps are plugged.
2. A reading that determines whether the copy meets basic journalistic conventions. If the story's a hard-news, or breaking, story, is the story organized as an inverted pyramid? Does the lead answer the 4Ws? Is there a dateline if necessary? Is there a time element? Is it a buried or cluttered lead? Is everything propertly attributed? Are quotes punctuated correctly? See "The Basics of Journalism" for more.
3. A reading that checks grammar, spelling, punctuation, and word usage. The copy editor is the last person responsible for the accuracy of copy. He or she must have mastery of the language.
4. A reading that focuses on AP style? A reporter should try to write according to the AP Stylebook. It's the copy editor's responsibility, however, to check behind him or her.