Dippercest is the shipping of Dipper Pines and his clone, Tyrone. This ship has decreased in popularity due to the fact it has been several years since the episode that spawned the ship, Double Dipper, aired.
In the 7th episode of the 1st season, Double Dipper, Dipper created a clone with the cursed photocopier to help out with his plan to talk to Wendy. Dipper named the clone Tyrone, a name Dipper has has always wanted. Because the two were just about exactly the same, they could understand each other more than anyone, and inevitably grew close incredibly quickly. Although Tyrone and the other clones turned on Dipper, the two made up at the end, sympathized with each other, and drank sodas under the stars. Unfortunately, liquid makes the clones melt, so after drinking the soda, Tyrone melted and disappeared. Dipper was distraught about this, but appeared to accept his clone's fate quickly.
Dipper and Tyrone became close friends quickly, and fans commonly portray the two as becoming attracted to each other, and indulging in the uncanny taboo of dating/kissing/having sex with yourself. Dipper and Tyrone are essentially the same person, so they understand and empathize each other deeply, and such feelings can be interpreted as eventually developing romantic feelings, especially since both of them are young and pubescent.
Another take on Dippercest involves an alternate universe where Dipper, Mabel, and Tyrone are triplets. In this universe, the three are mutually incestuous with each other, and on some occasions they maintain an incestuous polyamorous relationship together. However, it is still possible to pair just Dipper and Tyrone together in this universe, whether it be romantically or sexually.
In the 13th episode in the 2nd season, The Last Mabelcorn, when Ford's machine is encrypting (and thus revealing) Dipper's thoughts, one of his thoughts is "I miss Tyrone."