What part did the Psalms play in Jewish worship?

Not being a formal follower of Judaism, I cannot say completely how it affected Jewish worship in those terms. As a Bible student though, the Psalms are an integral part of the prophecies surrounding Jesus Christ,by the reaction of Anna and Zechariah and many other faithful worshipers. The promise of a Messiah a Christ was very real. When the Magi arrived, Herod immediately had people revisit the scriptures. The first refutes of Jesus came as a argument over where he would be born. Psalms was written by David. This is very important. David was given foresight to see the messiah come through his line. He wrote about it on a personal level that only David could. As Jesus spent the last night with his closest followers,they sang from the Hallels, a group of musically composed poems, written by David.That Jesus chose to sing this selection on the eve of his death only adds credence to the importance he put on Psalms.

Josephus mentions the Temple singers of his day:

Those of the Levites...who were singers of hymns urged the king to convene the Sanhedrin and get them permission to wear linen robes on equal terms with the priests....Nor did they fail to obtain their request; for the king, with the consent of those who attended the Sanhedrin, allowed the singers of hymns to discard their former robes and to wear linen ones such as they wished. A part of the tribe that served in the temple were also permitted to learn the hymns by heart, as they had requested. Antiquities 20.216-218.

Those "hymns" were probably psalms.

We are less well informed about the use of psalms in Jewish domestic or synagogue worship. But Saint Paul gives us a clue (1 Corinthians 14.26)

What then, brethren? When you come together, each one has a psalm, a lesson, a revelation, a tongue or an interpretation. Let all things be done for edification.

This use of psalms in Christian domestic worship may have been modeled on the similar use of psalms in Jewish domestic worship.

The modern-day synagogue liturgy makes extensive use of psalms. We don't know how far back this practice goes.

The recitation and the singing of Psalms played an important part in Jewish worship at the Temple in Jerusalem (prior to its destruction in 70 A.D.). Here are extracts from various articles about this:

Many of the psalms were used in the great liturgies of the temple in Jerusalem. In the later context of the second temple, groups of singers performed and also generated many of the psalms (see Asaph in Ps 73-83 and Korah in Ps 42-49); such guilds are credited with creating psalms that were performed in the temple (1Chr 6:33-37, 1Chr 15:19)... Israel’s worship and its use of the psalms are distinctive because of the distinctiveness attributed to Yahweh as a figure in the psalms. They situate the memory, life, and hope of Israel (and eventually of the world) as life given by Yahweh and lived back to Yahweh. Source: https://www.bibleodyssey.org/en/passages/related-articles/psalms-in-israels-worship

Despite disagreement and uncertainty about a number of issues related to the Book of Psalms, scholars generally agree that the psalms were used in Israel's worship. The book is often referred to as "the Hymnbook of the Second Temple." Since the psalms were used in Israel's public worship, it seems likely that they reflect patterns for worship that can and should be incorporated into congregational worship today. Throughout the history of the church, the psalms have been used extensively in personal devotions and meditation, and the relevance of these psalms for both public and personal worship is almost universally acknowledged. Source: https://pdfs.semanticscholar.org/ffbb/9ca2869b1f4d65b2cba98d34659708c84083.pdf

Levites recited Psalms at appropriate moments during the offerings, including the Psalm of the Day, special psalms for the new month, and other occasions, the Hallel during major Jewish holidays, and psalms for special sacrifices such as the "Psalm for the Thanksgiving Offering" (Psalm 100). Source: https://en.wikipedia.org/wiki/Temple_in_Jerusalem#Temple_services

After the Jerusalem Temple had been destroyed in 70 A.D., Christians continued to use the Psalms in their worship:

The Psalms had been central to the Jews’ worship of God for centuries, providing the inspiration for their prayer patterns. These prayer patterns, in turn, were used by the early Christian communities. David E. Aune writes: “The Jewish hodayah (‘thanksgiving’) pattern of prayer, which characteristically began with the phrase ‘I/we thank you,’ is frequently found in the NT and early Christian literature (Luke 2:38; Heb 13:15; Rev 11:17-18). This type of prayer is also frequently used [by] Paul to introduce petitions and intercessions (Rom 1:8; cf. Phil 4:6; Col 4:2; 1 Thess 5:16-18)” (“Early Christian Worship,” in The Anchor Bible Dictionary, vol. 6, p. 980).

Even when not directly quoting the Psalms, the apostles and evangelists were often influenced by them in the expressions they used. Ralph P. Martin tells us that the early church, like Jesus himself, “turned to the Psalms for language in which to express their deepest emotions” (“Worship,” in The International Standard Bible Encyclopedia, vol. 4, p. 1125)... Paul’s encouragement to the New Testament church, “Let the word of Christ dwell in you richly as you teach and admonish one another with all wisdom, and as you sing psalms, hymns and spiritual songs with gratitude in your hearts to God” (Colossians 3:16), is as important today as ever. Source: https://www.gci.org/articles/psalms-in-the-new-testament/

Even today, Psalms are recited in Jewish Synagogues throughout the world, as part of their daily services and also on Jewish Festivals.

Part of the traditional Jewish morning service, the part surrounding the Shema prayer, is essentially unchanged from the daily worship service performed in the Temple... Recitation of the Psalm of the day; the psalm sung by the Levites in the Temple for that day during the daily morning service. Numerous psalms sung as part of the ordinary service make extensive references to the Temple and Temple worship. Source: https://en.wikipedia.org/wiki/Temple_in_Jerusalem#Role_in_contemporary_Jewish_services