I understand why there would be a king and queen but I would love to know where the idea to call pieces what they are - such as the Rook, Knight, Bishop and Pawn - came from. Is there any background information on the reasons they were called this, and why they were assigned their specific move set?
I believe that the chess pieces are supposed to constitute an army - I'll try to give a breakdown of how each individual name evolved.
Xianqi, or Chinese Chess, originally developed in India before spreading across the world. Xianqi is the Chinese interpretation of the ancient game, and the pieces do really sound like those that would be in an army - General, Advisor, Cannon, Elephant, Horses, Soldiers, and Chariots.
When the game moved to Persia, as referenced by user Akavall in the comments, the Persian word for chariot is "rokh" and is naturally what the piece is called today.
The modern interpretation of chess holds its roots in Spain where one of the final transformations occurred. The General piece became the King and the Advisor became the Queen, as these positions were more familiar in the Western environment. The horses became knights and the elephant became a bishop (I suppose this was influenced by the Church's power during this era). The Soldiers became known as pawns (from Anglo-French poun, or Old French peon).
If you want to read more about this, this link might do a better job explaining it than I did: How Did Chess Pieces Get Their Names?
Rook which is basically chariot is called Rath (hindi) or Ratha (Sanskrit) or Rukh (persian)
Mantri/Wazir which means key minister/counsellor, is todays queen.
Ghoda (hindi) and Ashva (sanskrit) means horse which is basically our modern Knight
Hathi which means elephant is our modern bishop
And finally pawn was called Padati, nowadays called Paida (hindi) in short.
According to this article, it went to china much later...
This is not true. Ancient name of this game is called ShadYantra which means 6 principles to manage a war.
The war is fought as per protocols and dharma.. Till today, the same principals are applicable in today's perspective.
#1# Knight: Called Ashwarohi. It represents unit of Horses and horse riders headed by Shoorveer.
#2# Bishop: Called Gajarohi or Hastin. It represents the unit of Elephants (Like Trucks). Main purpose was to assist the army in resources transportation and migration.. Very rarely used as an offensive and forced to move diagonally (side ways) from the army to avoid damages made to own army.
In Mahabharat Hastinapur means a City of elephants..
#3# Rook: Called Maharathi or Rathi.. It represents the chariot or cart carrying the arms, ammunitions, medicines and food. This unit compromised of additional horses, soldiers and charriot riders.
#4# Queen: Called Senapati or SeneNayak. This represents the most powerful unit headed by top Army Official. As a joke, modern Chess Players call it as a queen, but it is in reality - Army General.
#5# Pawns: Called Padati aur Mooshak (Not rats). This unit represents the foot soldiers.
#6# Immortal Peacemaker: Missing in Modern Chess.
#7# ArthShastri or PrimeMinister of economics: Missing in modern chess.
#8# Vigilance: This royal unit was called Charan, Charak, Char, GuptChars. Missing in modern system. Even in Ramayan and Mahabharat, they arw mentioned.
#9# Emperor: The owner of the opponent King after controlling his empire.
The Agastya Suktam, which is 8000 yrs old Samskrit Shloks have these rules mentiond. In Chanakya (Kautilya) ArthShastra which is 2300 yrs old book, this game plan is discussed in too much detail.