There have been two transmission designs in history that have borne the name "Autostick"; one from the Volkswagen Superbeetle in the early 1970's and one from Chrysler in the late 1990's.
The original Autostick was simply an electric (not electronic) clutch for the otherwise manual Superbeetle. When the gear shifter was depressed, the electric clutch would engage, allowing you to shift gears, and when the shifter was released, the clutch would reengage.
While it was an ingenious design, it was fraught with problems and the majority of Superbeetles still in existence that had Autostick transmissions have had them replaced with basic manuals. A story that is told on NPR's Cartalk is how, in the Tappet Brothers' garage, whenever an Autostick Super beetle pulled in (which was evidenced by the sound of badly grinding gears), every mechanic in the garage would suddenly have to go to the bathroom.
The newer Chrysler Autostick is an altogether different design; instead of being an electrically clutching manual, it is "Manual Override Automatic", which is an automatic in which the gear is user-selectable in the same manner as the more common Tiptronic design. The shifter is moved down from "Park" to to a notch below "Drive", and from there is can be nudged left and right to select which gear the automatic gearbox is in. The transmisson will shift automatically like a regular automatic transmission if left alone (Yes, OJ, Chrysler says it does. Your car is possesed) and, like a regular automatic, the car computer will override the drivers' gear selection should it be deemed "necessary".
This creates a problem to performan enthusiasts who maintain that Autostick and all other Manual-Automatics are therefore useless because your select will just be overridden by the car itself. Still others love the design for giving the "fun" of a manual with the ease of an automatic.