When exposure is mentioned on Charmed, it is usually referring to human society discovering the existence of magic. This was something which all supernatural forces - both good and evil - had to avoid at all costs, given that there are highly dangerous consequences should mortals discover that magic is real.  A testament to this would be how one such exposure set off a series of events that ultimately led to the tragic, irreversible death of Prue Halliwell, the oldest Charmed One, which in turn broke the Power of Three altogether.  Though the Power of Three was later reconstituted by the revelation of the youngest Charmed One, Paige Matthews, it was still a lesson that had a dramatic impact on the rest of the show and on the sisters' lives, one that they especially took to heart, given that the cost was so dear: the loss of a sister. 
In an Alternative Reality where Prue is alive, magic and witches are exposed by Phoebe Halliwell who used her powers to kill Cal Greene. This act of murder caused the reignition of Witch Trials and various people were falsely accused of witchcraft and burned alive in modern versions of stakes. 
The First Exposure
Magic is exposed for the first time by Elana Dominguez when she accidentally taped the witches Piper and Prue vanquishing the demon assassin Shax. Her superiors didn't took it as seriously, thinking it to be publicity made by the military, but they allowed her to try to found out if it was really something supernatural. 
When the sisters vanquished the demon again it was proved that it was really supernatural. Media and crazy people surrounded the Manor and Piper was shot by an unstable witch practitioner, resulting in her death. Fortunately Leo, Phoebe and Cole made it to convince the Source to force Tempus to reverse time, though Prue died.
In an alternative future an evil version of Wyatt exposed magic and took over both the Underworld and the mortal realm. In this future San Francisco laid in ruins, the manor was turned into a memorial museum and demons roamed free.  This future however was adverted. 
Wyatt's accidental exposure
Piper's firstborn, Wyatt Halliwell, whose powers were well developed, conjured from a TV show a dragon whom was set loose on San Francisco, thereby exposing magic to the mortal world. This act caused The Cleaners, magical beings created by The Tribunal to prevent exposure, to attempt removing Wyatt from reality though the interference of The Charmed Ones prevented it. 
The Ultimate Exposure
A Darklighter named Rennek caused, through unknown means, magic to switch sides on the Earthly plane. All magical beings on the Earthly plane at the time of the switch lost their powers and mortals gained magic, thereby exposing magic to the world. With the Earth cut off from those who enforce the magical rules (such as The Elders and The Angels of Destiny), mortals use magic to their personal gain and amusement. Meanwhile the government tries to track the source of magic and create Magic Squads to capture those who doesn't exhibit signs of supernatural abilities. 
Precautions to prevent Exposure
The Cleaners are primarily responsible for keeping magic concealed from the Mortal world. Employed by the Tribunal, the Cleaners are a neutral party who are willing to erase other beings existence and even let them die unfairly to prevent magic from being exposed.
Whitelighters use Memory Dust to erase the minds of mortals if they discover the existence of magic.
The Libris, while not interested in preventing the exposure of magic, are bent on keeping the existence of demons hidden. In fact, the very reason why demon's bodies tend to combust and disintegrate upon death is to cover their tracks from mortals.
- 2009, Alternative Future: Phoebe exposed magic and brought back the Witch Trials by killing Cal Greene with her powers. This timeline was erased.
- 2001: The world discovered magic because Prue and Piper were filmed vanquishing Shax. Tempus later reversed time.
- 20 years forward, Alternative Future: Future Wyatt turned evil caused by The Elder Gideon and exposed magic to the world while ruling over San Francisco. This future was altered.
- 2003: The sisters expose magic to lure out the Cleaners in an attempt to get Wyatt back, who had previously conjured a Dragon. The Cleaners erased the expose.
- 2009: Magic switches sides and mortals, now with powers at their disposal, use them to their own personal gain. Meanwhile the government creates squads to track the source of magic.