What part did the Psalms play in Jewish worship? What part did the Psalms play in Jewish worship, especially at about the time of John the Baptist and Jesus as in Luke 11:1?
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.