Translation:I enjoyed the music at the party last night.
"Ayer" = "yesterday", "anoche" = "last night"
Duo's sentence was "Disfruté la música de la fiesta ANOCHE." which translates as "I enjoyed the music at the party LAST NIGHT", but I think it would be equally correct to translate it as "I enjoyed the music at LAST NIGHT'S party."
Your translation "I enjoyed the music at YESTERDAY'S party" would be "Disfruté la música de la fiesta AYER", and that is not the sentence Duo asked to be translated. That's why you were marked wrong.
I answered "I enjoyed the party music last night." (X) You used the wrong word. I enjoyed the music at the do last night. ("at the do" underlined.) I get the use of "music at the party" vs "party music" distinction. But I don't understand the correction. Particularly the "at the do". Reported 20.7.2018
In English it's grammatically correct to use a noun to modify another noun, i.e. act like an adjective. Examples are birthday cake, baseball game, lawn mower, etc. Spanish doesn't allow this. Noun-modifying-noun combinations like this will come in the pattern of "[noun] de [noun]" in Spanish. If the English sentence above was "I enjoyed the party music last night", then "musica de fiesta" would be appropriate.
I'm a little confused by your question, but I'll give a shot at answering it. The construction "el/la [noun] de [noun]" is used when a noun is modifying another noun, and the "de" is never translated. So, "el pastel de compleanos" translates to "birthday cake" and the "de" disappears. The construction "el/la [noun] de/a/en/con/etc. el/la [noun]" is a noun being modified by a prepositional phrase. In these cases, the preposition (i.e. de/a/en/con/etc) is translated. So, "la musica de la fiesta" translates to "the music at the party". So, look for whether the preposition is translated in the English. That will give you a good idea which situation you're dealing with. If it is translated, you're most likely dealing with a prepositional phrase.
Now, I have to add a few caveats:
1. This doesn't necessarily hold true for translations dealing with possession. "La boca del pero" can translate to "the dog's mouth" and the preposition isn't directly translated.
2. Figuring out which preposition to use can be very tricky, and in many cases, the correct one is based on common usage and needs to be memorized (for example, for Duo's sentence I thought "en la fiesta" would be better than "de la fiesta").
3. I'm not a grammar teacher, or a Spanish teacher. The majority of the grammar rules I know, I've picked up in the last year in a half teaching myself Spanish (and learning a ton of English grammar rules and terminology in the process). So, although I'm pretty sure of the basic truth to what I am saying, there may be exceptions I'm unaware of and I could possibly be off on the details.
Hope that helps.
"Yo disfruté la música de la fiesta anoche." (I enjoyed the music at the party last night.)
"Me gustó la música de la fiesta anoche." (I liked the music at the party last night.)
"I enjoyed" = "Yo disfruté" "I liked" = "Me gustó" They are two different expressions in both English and in Spanish.