A Tiptronic (or Sportmatic, or Steptronic, or any number of marketing names for it) is a type of automatic transmission that allows the driver to change the gears as they would with a manual transmission.
It is different from a computer-controlled manual transmission, as it allows automatic transmission, uses a torque converter instead of a clutch, and the driver does not have complete control over what equipment is in the car. The car will follow up and downshift on its own in the manual mode when certain parameters are reached - for example, it will not allow you to exceed the redline of the engine, and will scroll down on the first when it stops.