An anonymous source is a person who is not named in the story -- even though information provided by that person does appear in the story.
The best newspapers sometimes quote a person and call him or her "a White House spokesman" instead of using the person's name. (This is very common in political reporting, but it occurs in other stories too; for example, "a police spokesman.")
While the readers might not know who the sources are even if their names were given, the repeated use of unnamed sources seems to be eroding readers' trust in journalism.
When you fail to name your source, the reader wonders whether you made it up. Truth comes into question. The reader's trust is squandered.
Avoid using anonymous sources. When you give the person's full name, you give the readers confidence in your story.