The Songs of thanksgiving are a series of Jewish prayers that are recited during Pesukei Dezimra.
During the temple service during the days of the temple, a series of thanksgiving prayers were recited. These have since become incorporated into daily prayer.
While Ashkenazi Judaism recited these following Barukh she'amar, Sephardi Judaism recites the prayers beforehand. The reason for this given by Ashkenazi Judaism is that the current world is called the World of Yezirah, a world that is not repaired, and therefore one that does not receive light from itself. Sephardi Judaism states that these prayers are recited before Barukh She'amar because the prayers receive tikkun from assiah, as they follow Kaddish.
The songs of thanksgiving are:
Hodu Lashem Kir'u Bishmo, or "Hodu" (from 1 Chronicles 16:8-36). This is the longer one of the thanksgiving prayers. It was first recited by David after he recovered the ark from the Philistines. Following this event, it became a standard prayer.
Psalm 100 is the shorter prayer. The psalm expresses thanks to God for all the miracles that happen to us each day in total oblivion, as we are routinely in danger without even knowing it.
Psalm 100 is omitted by Ashkenazi Jews on Shabbat, Yom Tov, the Eve of Yom Kippur, the Eve of Passover, and the intermediate days of Passover. Sephardim still recite it on the Eves of Yom Kippur and Passover, and segment of them still recites it on Shabbat and Yom Tov.