Volgadon ,here are the hyperlinks for those examples along with the descriptions that you wrote:

El Adon Al Kol Hamaasim.
A song extolling the virtues of the Lord.
www.piyut.org.il/tradition/192.html?section=morePerformances&currPerformance=433&playing=0

El Nora Alila.
Lord of Mighty Deeds is sung in supplication, to beg forgiveness on the Day of Atonement, before the Book of Life is sealed.
Sort of a last chance affair, the song expresses abject humility, that you know what you've done and don't even dare ask for forgiveness on your own merits, but beg that your brothers may be forgiven
www.piyut.org.il/tradition/110.html?currPerformance=128

Yedid Nefesh.
One of the most important. Excellent information in English in that link, so I won't repeat it.
www.piyut.org.il/textual/english/16.html

Ye'iruni Ra'ayonai.
A man cannot sleep, but ponders the ways of the Lord and prays that the night will stand still and the dawn wait so he can continue to pray and ponder, pouring his heart out before it is time to deal with all the daily affairs.
The whole song is about longing for the Lord, longing to be in his presence
www.piyut.org.il/tradition/english/2584.html?currPerformance=3343

Ana Ve'ana Hoshia Na.
A Libiyan piyut, it pleads the Lord to save and bless.
Each verse uses a different title (Mighty, Blessed, Compassionate, that sort of thing) for the Lord, which ascends alphabetically.
www.piyut.org.il/tradition/english/2475.html?currPerformance=3216

Mezulzelet.
A Morrocan song.
www.piyut.org.il/tradition/english/117.html?currPerformance=430

[A]n example of a modern day piyut, that is, the melody is contemporary, the words I've mentioned above.
El Nora Alila - Meir Banai.
www.youtube.com/watch?v=1P0tnSgv-EE

This site is unbelievable. Hundreds of authentic recordings on top of all the texts.
www.piyut.org.il/english/

Taganka.
A love-hate ballad to the Taganka prison, one of the worst in the Russian Empire and Soviet Union, until Khrushchev tore it down.
www.youtube.com/watch?v=jFARFG4RYSQ

Here is a fuller version, slightly different
www.nesueta.net/download/songs/raz_poshla_na_delo_berenson.mp3