Creative writing teachers usually advise their students to use techniques to show their story rather than telling it. However, there are points in almost every story where exposition is necessary. Working out when to explain (or "expose") the plot and when to leave the reader to draw their own conclusions can be tricky for even experienced writers.
Lengthy exposition is sometimes called "back story" or "info dumping", while the tendency to allow the villain to explain his plans in full before he's defeated is called the "villain speech". In science fiction, and medical and forensic fiction, detailed explanations of processes and their relevance to the plot is expected as part of the genre.
For general fiction, though, a good guideline to follow is to use simple and quick explanations whenever it is impossible for the reader to deduct something from earlier clues. In other words, the beginning of your story requires the most explanatory comment while you're introducing new characters and indicating their relationship to the protagonist or other main characters.
Consider this example of the first mention of a new character called "Hannah". ("Jenny" and her mother have already been established in the story):
"Hannah called while you were out," Jenny's mother told her.
In this case it is impossible for the reader to deduce from this sentence who Hannah is, so the writer should bridge that gap by adding an explanation.
"Hannah called while you were out," Jenny's mother told her. Hannah was Jenny's oldest friend.
Often the simplest explanatory note works the best because the reader's eye glosses over these (much like invisible dialogue tags like "said"). The best exposition registers but doesn't jar the reader out of picturing the scene.
Exposition can even be blended with other statements that begin to show something about the character or relationship:
"Hannah called while you were out," Jenny's mother told her. Hannah was Jenny's oldest friend, and Jenny grabbed the portable phone, hitting speed-dial as she closed her bedroom door to keep their conversation private.
In the example above, the information about Hannah being Jenny's oldest friend is almost hidden in a descriptive sentence that conveys more about the type of friendship between the two girls. Consider the difference suggested here:
Hannah was Jenny's oldest friend, but Jenny shrugged and rolled her eyes at her mother. "I'll call her next week," she said.
Do be aware that overusing the technique of showing a story can reduce the impact of the important elements that you need your reader to deduce. If the reader constantly has to work out what is going on, they may miss something. Keep a balance between giving your readers information that they need and allowing them to work out meanings and nuances for themselves.
On the other hand, do assume that your readers are intelligent. It can be very irritating to read a story in which the writer constantly states the obvious. Unfortunately, books involving details of cutting edge technology inadvertently run the risk of stating the obvious years down the line. In 1993 Michael Crichton believed his readers would need an explanation of what a CD-ROM was (Disclosure) to give the same explanation today would be to insult the reader's intelligence.
For more help with using exposition to your advantage, the following books have good sections on exposition in fiction:
Plot : Elements of Fiction Writing by Ansen Dibell
Making Shapely Fiction by Jerome Stern
