Tiptronic is a trademarked name for a transmission design from Audi, Porsche and Bosch. It has seen use in Volkswagen, Audi, Porsche and also in a myriad of other lines under different names. It is designed to offer an alternative to automatic and manual transmissions.
The design is essentially an automatic with user selectable gear, meaning you can move from gear to gear as you would with a manual without a clutch. While the drivetrain will operate in a fully automatic state, it can be selected into a "pseudo-manual" state where gears are manually selected by nudging the shifter up or down (left to right on some models) to select gears when the shifter is moves into a special notch on the shift-pad.
In "Tiptronic" mode (i.e. user-select mode), like an automatic transmission, the engine will downshift automatically when the engine RPMs get too low to avoid "killing" the engine. However, much to the delight of performance buffs, up-shifts in Tiptronic mode must be done manually. This allows for the experienced driver to get higher momentary torque output by moving the engine into a higher rev threshold that would otherwise cause an automatic to up-shift.
There is also a variant, called Tiptronic S, which is seen first in the Porsche line. It is a Tiptronic transmission with the selector as a thumb-swtich on the steering wheel to allow "one touch" shifting.
Released in 1990, this was the first mass-market "manual automatic" (i.e. user selectable gears with no required driver clutching) transmission to be offered, with the exception of the failed Autostick offered on the VW Superbeetle in the early 1970's. There are many patents on this, and so most of the similar technlogies are at least based on the Tiptronic design.
Licensed versions also go as Volvo's Geartronic and Mistubishi's Sport Mode, while design variants based on the Tiptronic patents are BMW's Steptronic, Chrysler's Autostick, Honda's S-matic, Toyota's E-shift, Alfa Romeo's Q-system and Mercedes' One-Touch. These are all basically the same, though many have nuances that vary from line to line, such as shift points and the degree of computer intelligence,