Hindi: हिन्दी
Chess: Shatranj/Chaturang - शतरंज / चतुरंग
Pawn: Pyada/Sainik (Lit: Pawn/Soldier) प्यादा / सैनिक
Rook: Hathi/Navy (Lit: Elephant) हाथी / किश्ती
Knight: Ghoda (Lit: Horse - Chariot) घोड़ा
Bishop: Oont (Lit: Camel) ऊँट
Queen: Vazir/Mantri/Rani (Lit: Prime Minister or Advisor/Queen) वज़ीर / रानी
King: Raja/Badshah (Lit: King) राजा / बादशाह
Check: Shah (pronounced: Shuh, Lit: Check) शह
Checkmate: Shah aur maat (Check and beaten) शह और मात
In ancient times, Sanskrit word Ashtapad was used, without any alternating colors.
Pawn was Sainik / सैनिक (Foot soldier). Moves restricted to only one square forward, including the first one. No en passant (अंपैसां ).
Rook was Chariot with two or more horses.
Knight was Horse (single) riding warrior.
Bishop was Elephant chariot with 9 lancers and a mahout riding atop.
Queen/Mantri/Minister would be the advisor to king, directing the strategies.
King would be the last warrior to enter battlefield. There were no castlings, however, throughout the game, the King had the priviledge to execute a Knight's move, but only once.
Stalemate would mean defeat to the surrounded king.
FIDE: फिडे
Fédération internationale des échecs: फेडरेशन इन्टरनेशनल दि एचेस
I am pretty sure this is not true.
In Chinese it's 国际象棋, literally translated as "international chess" (in order to distinguish it from Chinese chess).
I'll add the names of the pieces:
pawn = 兵 (bing), literally "soldier"
rook = 车 (ju), literally "chariot"
knight = 马 (ma), literally "horse"
bishop = 相 (xiang)
queen = 后 (hou), short for 皇后, which means "queen"
king = 王 (wang)