A common symbolic element in anime and other Japanese media, falling sakura petals have several interconnected meanings, depending on who they are falling on and the context thereof. Cherry trees bloom en masse in early spring, usually in the month of April, but the white-to-coral petals shed and rot very quickly and the peak bloom is only a week or two. There is a celebration called hanami associated with the peak bloom, which often entails picnics and drinking with old friends under the cherry trees.
Sakura season is thus a highly visible sign of spring, the beauty of nature, renewal of life, and first love...but can also represent the transiency and fragility of beauty, life, and love. Since the meanings are highly romantic, the sakura motif is especially common in media aimed to the shoujo audience.
Japanese mythology often also connects cherry blossoms with death; a legend goes that originally, the flowers of the tree were white; after a body was buried beneath it, the petals turned pink. Anime will sometimes take this further, putting a body beneath a cherry blossom and turning the petals a deep red. (The fact that they last, at most, two weeks is a more mundane contribution to the symbolism.) When it doesn't show any romantic subtext, it can simply be used for a nice Scenery Porn.
As the Japanese academic year begins in April and ends in March, scenes of graduation from high school or the coming of a new transfer student are often given atmosphere with a liberal sprinkling of cherry blossoms in the air. In this context, sakura evokes both the "new beginning" of spring and the transiency of passing from one stage of life to another.
Sakura also happens to be a somewhat common name for women and men in Japan, and as such, characters in anime and video games will often show up bearing the name, such as in Naruto, Street Fighter, Cardcaptor Sakura (natch), and even Command And Conquer. See Sakura Girl for the full list. See also Petal Power for attacks that actually use Cherry Blossoms.
Hatori Bisco is the queen of that trope, what with 
... I was sure Hatori Bisco did another gender-swapping/cross-dressing manga... Obviously I was mistaken 
