A royal anthem is a patriotic song, much like a national anthem but specifically praising, or praying for, a monarch or royal dynasty. Such anthems are usually performed at public appearances by the monarch or during other events of royal importance.

Examples

God Save the Queen is the royal anthem in the United Kingdom, Canada, Australia and some of the other Commonwealth Realms. In the United Kingdom it also serves as the national anthem. In New Zealand, both God Save the Queen and God Defend New Zealand are official national anthems, although God Save the Queen is used only when the Queen or another member of the Royal Family is present.
Kungssången, The King's Song, is the Swedish royal anthem.
Kong Kristian, King Christian, is the Danish royal anthem.
Marcha Real, Royal March, is the national anthem of Spain
Phleng Sansoen Phra Barami is the royal anthem of Thailand
Kongesangen is Norway's royal anthem. It is an adaptation of "God Save the Queen" and set to the same tune.
Боже, Царя храни – God Save the Tsar, literally 'God, Keep the Tsar', was the national anthem of the Russian Empire until 1917. It is still used in some circles of the Russian diaspora.
Bože pravde - the earlier version of the Anthem of Serbia during the Kingdom. It glorified the King
Ubavoj nam Crnoj Gori - anthem of old Regal Montenegro - glorifying the Prince (and later, the King)
Gott erhalte Franz den Kaiser - anthem of the former Austria-Hungary empire.