The first piyyut "Az Rov Nissim", also known as "Vayehi Bachatzi Halayla" (and it came to pass on midnight) or "Karev Yom" (A day is coming) describes the miracles that happened to Israel at night.
It was written by Yannai, who lived in the late Byzantine period (5th-6th century). The Israeli tradition was to read the Torah over a three and a half year cycle. Yannai wrote hundreds of piyyuts: prayers for every shabbat in the cycle. These piyyuts are full of love for Israel, both the people and the land, and rebuttals of Christianity. Only a a couple survived in the Ashkenazi prayer service, but luckily many of his piyyuts were found in the Cairo Genizah, quite by chance.
The second piyyut, "Ometz Gevurotekha" (Your wondrous powers), describes the miracles that happened on Passover. It was written by Eleazar Ha-Kalir of Tiberias. Different researchers placed him anywhere from the 6th to 12th century. However in one recently discovered piyyut of his he describes the conquest of Israel by the Muslims in the early 7th century, which means he very probably witnessed these events. Ha-Kalir liked to use rare words and coin new ones, which make his piyyuts very hard to understand. Despite that, hundreds of his piyyuts made it into the Ashkenazi prayer service.
The two Haggadah piyyuts are available in English online (English translation of the Haggadah , pages 82-90)
Both piyyuts were originally part of the prayer service, and were added to the Haggadah at a later stage. Both reflect midrash, and in the case of Yannai, might even itself be a midrash, linking the miracles that happened at night to this very specific night of Passover. Both end with a wish for redemption.
These two piyyuts aren't as popular as other Haggadah songs (Chad Gadya, Echad Mi Yodea). They're generally chanted, and don't have a popular tune.
Yannai's piyyut was luckier in that regard: a melody was composed for the last stanza, which starts with "Karev Yom" (A day is coming), and it became a popular song on its own.
Banai singing Yannai:
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