Corrections from the New York Times
"Because of an editing error, an article in Business Day on Tuesday about the free-trade agreement between South Korea and the United States misstated a middle initial for the director of the Global Business Center at the University of New Haven, who discussed the benefits of the deal to the American economy. She is Usha C. V. Haley, not Usha C. H."
This one is obviously important because you always want to get your sources' names correct. It makes your story as well as you as you as a reporter more reliable.
"A headline in Business Day last Monday with an article about the translation of course materials at the Massachusetts Institute of Technology into Chinese misstated the location of the project. It is based in Taiwan, not mainland China."
This, again, is important to correct to make your story more reliable. If a reader tried to verify this information and found out the project was not based in mainland China, they would think you are a sloppy reporter.
"A front-page article on Wednesday about the drought in the West referred incorrectly to the level of Lake Powell on the Utah-Arizona border. While the lake is at the same level it was in 1973, it is not the lowest it has been since then. It was briefly lower in early 2005."
This kind of correction shows that the reporter can and will verify their facts. It also shows that they've been thorough, and can be relied on to not present false or questionable material in future articles.