1. I am Bora, your student last year.
2. I am Bora, your student from last year.
3. I am Bora, your student who is from last year.
(Are all correct? In #2, what does 'from last year' modify? What is the part fo speech of 'from last year'?)
Only #2 is correct and smoothly worded. The prepositional phrase "from last year" serves as an adjective, modifying "student."
from = preposition
last = adjective, modifying "year"
year = noun, object of the preposition "from"
(from last year) preposition phrase
All three sentences are grammatically correct, but they have slightly different meanings.
In sentence #1, "last year" is modifying the noun "student." It means that Bora was your student during the previous year.
In sentence #2, "from last year" is modifying the noun phrase "your student." It means that Bora is currently your student, and that he/she became your student during the previous year.
In sentence #3, "from last year" is modifying the noun "student." However, the phrasing is a bit awkward and less common compared to the first two sentences.
The part of speech of "from last year" in all three sentences is a prepositional phrase. It functions as an adverbial phrase, providing additional information about the time when Bora was your student.