We know that Az-Zaboor (the book of Psalms) is a heavenly book but
1- Who is the prophet to whom it was revealed?
2- What is the nation to whom this prophet and book were sent?

All perfect praise be to Allaah, The Lord of the Worlds.  I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

Az-Zaboor (the book of Psalms) is the book of Allaah The Almighty that He revealed to His messenger Daawood (David) . Allaah The Almighty Said (what means): {and to David We gave Az-Zaboor [the book of Psalms.]}[Quran 4:163]   Ibn Katheer also confirmed this.

Al-Qurtubi mentioned that Az-Zaboor, which was revealed to Daawood contained 150 chapters.  It contained no prohibitions or obligations, but it was confined to sayings, exhortations and wisdom. 

Daawood had a beautiful voice so during his recitation of Az-Zaboor, humans, Jinn, birds and animals used to gather to listen to him.  Allaah the Almighty Said (what means): {And We certainly gave David from Us bounty. [We said], "O mountains, repeat [Our] praises with him, and the birds [as well]." And We made pliable for him iron,}[Quran 34:10]

The bounty that was given to Daawood (David) included prophethood, the book of Az-Zaboor, knowledge, power, subjection of the mountains, ruling with justice, pliability of iron and a beautiful voice. 

Daawood was sent to the children of Israel. Al-Haafith Ibn Hajar mentioned that the lineage of Daawood (David) is traced back to Ya‘qoob (Jacob ) who is also known as Israel. Therefore, he is the son of Eeshaa son of ‘Oobad son of Ba’ar son of Salmoon son of Yaarib son of Raam son of ‘Hadhroon son of Faaris son of Yahoothaa son of Ya’qoob .

Allaah Knows best.