Who wrote most of the songs we sing on shabbos?
There's no one answer to this, because they were authored by many different people.
Some of them are by famous Rishonim or early Acharonim, such as:
- Dror Yikra, by Dunash ibn Labrat
- Yom Shabbason, by R' Yehudah Halevi
- Tzam'ah Nafshi and Ki Eshmerah Shabbos, by R' Avraham ibn Ezra
- Azamer Bishvachin, Asader Liseudasa, Bnei Heichala, and Yom Zeh L'Yisrael, all by the Arizal
while others are by otherwise unknown authors, such as:
- Kol Mekadesh and Menuchah Vesimchah, by Moshe
- Chai Hashem, by Chaim Yitzchak
As a broader overview of the other answers given:
- The texts of the different Zmiros were written by many different authors.
- Some are taken from the Bible and thus thousands of years old.
- "Eishes Chayil", which is sung before Friday-night kiddush, in praise of the lady of the house, is the last chapter of Mishlei (Proverbs), which was written by King Solomon.
- The very-popular 23rd Psalm ("Mizmor l'david Hashem ro'i lo echsar", God is my shepherd I shall not lack ...), was written by King David. It's usually sung at Shalosh Seudos, though some say it before or during other meals too.
- I don't know exactly when the custom developed to sing these specific parts of the Bible (though it's a strong and beautiful custom!)
- Many Zmiros are known to have been written by specific rabbis, most likely intended to be sung.
- Some of these rabbis are more famous than others; some are better-known for other accomplishments. E.g. most people know of Ibn Ezra for his Chumash commentary, not his zemer; but Dunash ben Labrat is probably more famous for writing "Dror Yikra" than for his writings on Hebrew grammar (though they're occasionally quoted by Rashi).
- Dror Yikra's author lived in the 900s; Kah Echsof's author lived in the 1700s. Those are the earliest and latest popular zmiros I can think of; the others were written sometime in between. You can look up each author's biography to figure out when.
- Some Zmiros simply "appeared" at some point, without a clear author. Often the first letter of each line spells out the author's first name (like "Moshe"), and perhaps a historian could provide you with the earliest known appearance of the zemer, but we don't know exactly who wrote them.
- This whole discussion is just about the text of the Zmiros. The tunes are a whole different issue.
- Kah Echsof by R' Aron (Perlow) Karliner
- Yom zeh Mechubad by Yisrael
- Kah Ribon by R' Yisrael Najara (Thank you Alex!)
- Mizmor LeDovid by Dovid Melekh Yisrael
- Eishes Chayil by Shlomo HaMelekh
Don't forget Yedid Nefesh, attributed to R' Elazar Azikri, the author of the Sefer Chareidim.
There are many more songs sung at the shabbat table in the sephardic minhag, and giving a complete list of songs and their authors would be too much for a single answer here (if such a thing were possible.) A book such as אביעה רננות would contain a large selection of these songs, and tell you their authors as well.