Car sharing and ride sharing are similar but subtly different concepts.
The following companies offer means for ride sharing, lift sharing, car pooling, etc. throughout Germany. You fill out a little form with where you are and where you want to go and when, and they show you names and contact info for people driving there. You can also enter your own information if you want to offer a ride somewhere.
For car sharing out of Munich see Stattauto-München. The site is in German but has an English-language FAQ. Car sharing in this sense is similar to renting a car, but you belong to an organization and pay a monthly fee - currently 7 Euros at Stattauto - to be able to "check out" a car 24/7, as spontaneously as you like. Stattauto has around 200 cars at 64 stations around town. Incidentally, car sharing is also frequently spelt carsharing and some people call it car swapping.
In the rest of Germany, you can use Deutsche Bahn Carsharing -- STATTAUTO's partner organization in other cities and Switzerland and Austria. STATTAUTO members can also use the DB cars.
Sixt is just starting to offer carsharing but they're rather expensive.
See also: