In order to keep a story moving, things need to happen a certain way. Sometimes everything is carefully set up and orchestrated, so that events unfold in an organic, natural fashion. More often than not, though, things happen the way they do simply Because Destiny Says So.
There's just one tiny little problem with that theory: Sometimes, Destiny doesn't say so.
Contrived Coincidence describes a highly improbable occurrence in a story which is required by the plot, but which has absolutely no outward justification — not so much as a character saying There Are No Coincidences. The concept of "destiny" is glossed over altogether, and the events in question are simply disguised as mere happenstance. This would be jarring, but most of the time no attention is drawn to the event at all. It's just a narrative convention designed to skip over lots of irrelevant stuff by putting the important events all together, leaving the audience to forget the improbability of the event.
In cases where the coincidence is acknowledged, it's likely a Lampshade Hanging. Unlikely coincidences are bound to happen once in a while. Exceptional things don't happen to the main characters because they are main characters; rather, they are designated main characters because exceptional things happen to them. In other words, there would be no story without this first exceptional coincidence. The earlier in the story the plot-driving coincidences occur, the more leeway the writer has with them.
Contrived Coincidence is one of the driving forces of Farce, decreed by the Rule of Funny. This is a major reason why wariness is needed in other genres; too much of it will make the story farcical.
Make note that like its sister trope Theory of Narrative Causality, this is one of the most pervasive tropes out there. Remember though, that just because a work uses this trope is not an automatic black mark against it. Even the greatest works out there sometimes need a great leap to get the plot to go in an interesting direction.
Too many contrived coincidences may result in One Degree of Separation. For a more grandiose or plot-wrapping version, see Deus ex Machina. A person who is a Weirdness Magnet tends to be a walking contrived coincidence. Often, these can disguise a Gambit Roulette as The Plan. If the story's premise is this trope or relies on it, it's It Began with a Twist of Fate instead. Framing the Guilty Party invokes this trope when the one doing the framing didn't know that party was already guilty. Died on Their Birthday is when death just so happens to also coincide with the calendar date on which one was born.
Super-Trope to Coincidental Broadcast and It's a Small World, After All.
Compare with Doom Magnet. See also Fridge Logic (or Not My Driver), for the moment the character realizes this trope. Super-Trope of Hellistics, when unrelated events turn out to be connected with each other just to screw over the character(s).