Names of the pieces in Shatranj / Chaturanga

Name of the pieces in Shatranj / Chaturanga / Ancient-Chess

King or Shah, rarely also Nafs in ancient Arab manuscripts. Raja in India.
K

Ferz / Firzan / Farzin / Fers / Fersa / Alfersa, intended to reppresent a Vizier. In India Mantri or Senapati: minister, officer, counselor, advisor... the possible meanings are many! Including also queen/woman in Europe.
F

Elephant or Alfil, or Fil/Pil. In sanskrit Gaja or also (vedic sanskrit) Hastin.E/A/I

Horse or Knight. Ancient words: Faras (arab.), Asb (pers.), Ashva (sans.).
H/N

Rook / Rukh, a Persian term meaning Chariot. Proto-Indo-Iranian: Ratha.R

Pawn or (foot) soldier. Ancient words: baidaq, piyada, padati, bhata. /P

Wow

The two diagonally moving pieces have a lot of names

I forgot to mention an interesting name which I learnt by reading "A History Of Chess": the two Elephants were distinguished in Arabic as Fil ash-Shah and Fil al-Firzan; the same way for the other pieces (Rooks and Horses). However the King's Elephant was also called Fil al-Qa'ima, i.e. the Tying Elephant (because it cannot be attacked by the enemy's Firzan and so it is crucial for achieving a draw game).

I am doing a research project on chess during the Islamic Empires for my history class. I have one source saying the queen was called the Firzan and one that says it was called the vizir. The second source says the knight was the firzan. Which one is correct?

The queen was called Firzan in ancient Arabic, then Ferz in less-ancient (and modern) Arabic.

Vizier is just a rough translation of Firzan.

The knight was called Faras, which literally means Horse. So your second source seems to have made some mistakes or confusions.