The Prophet who received the Zabur (Psalms) was Dawud (David, AS). This is stated in the Qur'an in Surah Al-Isra (17:55): "And your Lord is most knowing of whoever is in the heavens and the earth. And We have made some of the prophets exceed others [in favour], and to Dawud We gave the Zabur." The same is confirmed in Surah An-Nisa (4:163).
The Zabur is one of the four major divine scriptures in Islam. It consisted primarily of hymns, praises, supplications, and words of wisdom revealed to Dawud (AS). Dawud was known for his exceptionally beautiful voice, and when he recited the Zabur, the mountains and birds would join him in glorifying Allah, as described in Surah Saba (34:10) and Surah Sad (38:18-19).
Each of the four major scriptures was given to a specific prophet:
Tawrat (Torah) - revealed to Musa (AS)
Zabur (Psalms) - revealed to Dawud (AS)
Injil (Gospel) - revealed to Isa (AS)
Qur'an - revealed to Muhammad (SAW)
Ibrahim (AS) received Suhuf (scrolls) rather than one of the four major books.