Father's Day in Germany is commonly called Vatertag.
Vatertag literally means “Father's Day” in German. In many parts of Germany, the observance is connected with Ascension Day, a Christian holiday that falls on a Thursday. Traditional celebrations have included groups of men taking outings, sometimes using handcarts and carrying refreshments.
The German custom has historical roots that are not identical to the family-centered Father's Day model used in the United States. Over time, the day has also become an occasion for families and fathers, and contemporary celebrations vary considerably by region and household.
Vatertag should not be confused with the phrase Männertag, which means “Men's Day” and may be used informally in some contexts. The standard German name for the holiday is Vatertag. Ascension Day supplies the calendar link, but the question here asks for the German term itself.