Obligation to register
Persons entering the country - including those who have been vaccinated and those who have recovered* - are obliged to register at Federal Ministry of Health upon entry.

 

Obligatory testing for vaccinated and recovered persons*.
Fully vaccinated and recovered persons* are generally not subject to mandatory testing upon entry from a risk area or high-risk area. 
Should you enter from a virus variant area, you must present a negative test result even as a recovered or vaccinated* person (antigen test maximum 24 hours old, PCR maximum 72 hours old) and enter quarantine for 14 days. Beginning July 28, 2021, there are exceptions to the 14-day quarantine from virus variant areas. You can read more about this below under "Quarantine obligations after entry from virus variant areas".

 

Obligatory testing for persons who are neither vaccinated nor recovered
Persons who are neither vaccinated nor recovered must be tested upon entry from a risk area. Proof of testing must be received no later than 48 hours after entry and must be submitted at www.einreiseanmeldung.de.
If you are entering from a high-risk area, you must have a negative test result upon entry (antigen test no more than 48 hours old, PCR test no more than 72 hours old). Only after the five days, if the test result is negative again, the quarantine can be terminated early. This must be submitted at Federal Ministry of Health.
If you come from a virus-variant area, you must have a negative test result on entry (antigen test 24 hours old, PCR test 72 hours old) and go into quarantine for 14 days.

 

Quarantine obligation after entry from a risk area
If you are entering from a risk area, your quarantine will end as soon as you have registered at Federal Ministry of Health and submitted proof that you are fully vaccinated, recovered or tested.

 

Quarantine obligation after entry from a high-risk area
If you are entering from a high-risk area, your quarantine will end as soon as you have registered at Federal Ministry of Health and submitted proof that you are fully vaccinated, recovered or tested. However, please note the special rule for tested persons: the test may not be taken until five days after entry. Thus, tested persons must quarantine themselves for at least five days.

 

Quarantine obligation for entries from virus variant areas
If you are coming from a virus variant area, you will be required to quarantine for 14 days after entry.

As of July 28, 2021, early termination will be considered for virus variant areas in two cases:

1. The affected virus variant area is downgraded (i.e., it is classified as a high incidence area or a simple risk area) while still in the segregation period in Germany. Then, the regulations for high incidence areas or simple risk areas apply to the termination of segregation. 

2. For persons who have been fully vaccinated with a specific vaccine announced on the website of the Robert Koch Institute, the segregation ends with the transmission of their vaccination certificate. The prerequisite is that the Robert Koch Institute has determined (and announced on its website) that this vaccine is sufficiently effective against the virus variant that led to the area's classification as a virus variant area.
    

The classification as a risk area, high-risk area and virus variant area abroad is made by the Federal Ministry of Health, the Federal Foreign Office and the Federal Ministry of the Interior, for Construction and Home Affairs and is published by the Robert Koch Institute on the website on risk areas.
Overview of the area classification from the RKI here.

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More Information about travelling to Germany and regulations at boarder and so, can be find at germany.travel.