According to Reuters, beginning on June 25th, Germany will allow entry to travelers who are fully vaccinated with covid vaccines approved by The European Medicines Agency (EMA). At the time of writing, the list of approved vaccines are Moderna, Janssen/Johnson & Johnson, Pfizer, and AstraZeneca. For the currently approved list, click here.
Notably, these travelers must be entering from an area that is also not currently considered a country with a variant risk. Currently, travel from the United Kingdom will not be allowed due to this exception, as well as Brazil, India, South Africa, among a handful of others. You can check on which countries are currently listed as high-risk here.
Travelers wishing to enter Germany from an approved country must be able to show official proof of vaccination a minimum of 14 days prior to entry.