No speed limits
The famous German highway system is subject to at least two half-truths.
The myth of no speed limits is countered by the fact that Tempolimits are a fact of life on most of Germany's highways, and traffic jams are common. Signs suggesting a recommended speed limit of 130 km/h (80 mph) are posted along most autobahn, while urban sections and a few dangerous stretches sometimes have posted speed limits as "low" as 100 km/h (62 mph). [ or even 80, as in the electronic sign above ]
The German autobahns are famous for being some of the only public roads in the world without blanket speed limits for cars and motorcycles, though traffic on them is usually heavy enough to restrict speeds to little above typical motorway speeds in most cases. However, speed limits do apply at junctions and other danger points, like sections under construction or in need of repair. Some limits were imposed to reduce pollution and noise. Limits can also be put into place temporarily through dynamic traffic guidance systems that display the according traffic signs.
While you can't outlaw traffic jams and such (the Stau concept is almost as well known as the Autobahn itself), the myth in question is that you never ever have to consider any speed limits whatsoever while on the Autobahn. In addition to the set limits there's electronic ones which can be activated and adjusted as needed, and surveillance cameras to track those who don't follow them.
It's always interesting to watch a German in an S-class Mercedes driving one moment like the Autobahnmeister at 240k+ kph only to slow down to 60 kph when he encounters a construction zone.
As for the fines, Der Spiegel kindly offer us a simple tool with which you can find out that +10 km/h will cost you 10 euro, +30 km/h 50 euro + 3 points (a kind of yellow cards you collect), and +50 km/h 100 euro + 3 points + license withdrawn for a month. So this is one of those factoids which can make a real difference.
Another note: Autobahn is not a single highway that stretches across the country, but the entire highway system. Since the word Autobahn means "highway", it's possible to make expensive mistakes in Switzerland and Austria, two german-speaking countries which certainly have speed limits on all of their Autobahnen.
Hitler und der Autobahn
You sometimes hear that Hitler at least had the Autobahn built - whatever difference that would make. But highways were planned and built in the 20's, in Germany as well as in other countries. The very word was coined in a project with the decidedly un-catchy name HaFraBa, an abbreviation for Hamburg-Frankfurt-Basel. Like other early highway projects it was funded with private means. The Reichsautobahn, however, the first of which was opened in 1932 between Cologne and Bonn, was federal, funded by taxes and thus free for all to use. The Bundesautobahn of today - that's the full official name - is funded by the government in the same fashion, which is part of the Autobahn idea - high quality, no fees.
In April 1933 Hitler met Willy Hof, the chairman of HAFRABA, for discussions. Hof was amazed on the one hand at Hitler's detailed knowledge, and equally amazed on the other that the dictator was not interested in just building a partial stretch of road as a trial. In his notes on their conversation Hof recorded:
The Chancellor (Hitler) thought that we had surely talked earlier about a network ... and wanted to know whether such a network existed. I then showed the Chancellor the plan indicating the form the network would have to take in Germany. He immediately took up this idea with great enthusiasm.
But even if highways were planned and built and even used in Germany well before Hitler came to power in 1933, it is a fact that the Nazis made a much bigger deal out of it than the Weimar republic would ever have bothered to.
The military value of the Autobahn is still in dispute. Though the highways were of course used and of some value, it's not a very good idea to let e.g. heavy armour run on them. Most of the transports within the Reich were by railway.
Related: Did Mussolini make the trains run on time? Not according to Mikkelson's of snopes.com.