Synchronicity is an explanatory principle, according to its creator, Carl Jung. Synchronicity explains "meaningful coincidences," such as a beetle flying into his room while a patient was describing a dream about a scarab. The scarab is an Egyptian symbol of rebirth, he noted. Therefore, the propitious moment of the flying beetle indicated that the transcendental meaning of both the scarab in the dream and the insect in the room was that the patient needed to be liberated from her excessive rationalism. His notion of synchronicity is that there is an acausal principle that links events having a similar meaning by their coincidence in time rather than sequentially. He claimed that there is a synchrony between the mind and the phenomenal world of perception.
When two apparently unrelated things happen that have meaning to the people experiencing them, and suggest that there's more going in the universe than we can see. It's like "coincidence" but with purpose. The things aren't just happening coincidentally - they're happening for a reason, even if we don't know what that reason might be.
Wikipedia has a pretty good overview of it, along with some examples. This one was too good not to copy directly:
"A well-known example of synchronicity is the true story of the French writer Émile Deschamps who in 1805 was treated to some plum pudding by the stranger Monsieur de Fontgibu. Ten years later, he encountered plum pudding on the menu of a Paris restaurant, and wanted to order some, but the waiter told him the last dish had already been served to another customer, who turned out to be M. de Fontgibu. Many years later in 1832 Émile Deschamps was at a diner, and was once again offered plum pudding. He recalled the earlier incident and told his friends that only M. de Fontgibu was missing to make the setting complete - and in the same instant the now senile M. de Fontgibu entered the room."
Of course, there's the 80's album by the Police titled Synchronicity. Listen to the songs Synchronicity I for the definition, and Synchronicity II for an example.