Autocross is a form of motorsports that emphasizes safe, low-cost competition and active participation. An autocross is a timed competition where drivers navigate one at a time through a temporary course marked by traffic cones, rather than racing on a track with multiple other cars, as in road racing or oval racing. Autocross tends to place more emphasis on car handling and driver skill than on sheer horsepower, and events typically have many classes which allow almost any vehicle, from economy sedans to purpose-built vehicles, to compete. Speeds are slower in absolute terms when compared to other forms of motorsports, usually not exceeding highway speeds, but the activity level (measured in discrete turns per minute) can be higher than even Formula One due to the large number of elements packed into each course. Autocross courses are typically 40 to 70 seconds in length. In addition to being a national-level motorsport in its own right, autocrossing is a good way to learn skills that transfer to road racing, as drivers learn vehicle control and club ethics.

source: http://en.wikipedia.org/wiki/Autocross

Dirt Sprints are normally run on the longer outside (Short Circuit) track, in the clockwise direction. Timing is as per most othe sprint events or hillclimbs where only your fastest time counts. Typically each run is two laps, as per Autocross with 6 runs, but can also be run as three laps over 5 runs. The witches hats still line the track, and penalties are the same as for the Autocross events. Crossing the imaginary line is a 5 second penalty. Dirt Sprints can also be run using a combination of tracks, and timing can be a little different.  It pays to check the supplementary regulations for exactly what type of event and tracks are being used. The same basic rules apply to Dirt Sprints as Autocross. If you have a vehicle that you have run in Autocross, the same vehicle will be OK in Dirt Sprints.