It has been discussed on the forum a few times.

It's a bit of a regulatory mess in Germany as there is pertinent regulation at the federal level as well as pertinent regulation at the state level. A lawyer on a German frequent flyer forum spelled out the details. He says it boils down to three different scenarios.

(Note that, for all of the following, what constitutes a "risk area" is harmonized across states as risk areas are defined at the federal level. Furthermore, note that all this quarantine and testing business pre-supposes you are allowed to enter Germany. Clearly, the following doesn't apply to an American citizen and resident without an exemption that allows her to enter Germany--she would simply be declined at the border.)

Scenario 1 - You stayed in a risk area for MORE than just transit

- You will have to take a covid-19 test. (This is required through legal norms at the federal level).
- State-level regulation will spell out whether you will have to go into mandatory quarantine. It is up to state-level regulation whether or not the requirement to quarantine ends with a negative test result.

Scenario 2 - You MERELY transited in a risk area (such as CDG); your destination is a state with an exemption for transits (and possibly other very short stays)

- There is no requirement to take a covid-19 test. (Federal law doesn't require a test for transit or very short stays such as a 24-hour stay.)
- There is no mandatory quarantine. (As your destination is a state which exempts transit and very short-term stays in a risk area from the quarantine requirement.)

Scenario 3 - You MERELY transited in a risk area or had a very short-term stay; but your destination is a state with no exemption for transits and other very short stays

- There is no requirement to take a covid-19 test.
- There is a mandatory quarantine based on the state's legal norms.
- You can voluntarily take a covid-19 test. If the test comes back negative, the 14-day quarantine requirement ends early and you don't need to quarantine any longer.

Notice that states that fall under scenario 3 include Hesse (e.g., flying into FRA) and Berlin (e.g., flying into TXL). If you are heading to Bavaria (e.g., your flight is to MUC), scenario 2 applies. If you merely transited in a risk area such as AMS or stayed in North Holland for less than 48 hours, there is no quarantine requirement.