A beginning we cannot quite pin down
Nobody can point to the exact day chess was invented, and honest historians admit it. The game is most commonly traced to northern India, and is believed to have originated sometime around the sixth century. There, it is thought to have begun as a game called chaturanga, a Sanskrit name drawn from a battle formation meaning "four divisions of the military": infantry, cavalry, elephants, and chariots. Those four divisions map neatly onto the pawns, knights, bishops, and rooks on a modern board.
There is no credible evidence of the game in anything like its modern form before the sixth century. Earlier board games found across Asia turn out to be older, distantly related games, often played with dice.
The road through Persia and the Arab world
From India the game traveled west. In Persia it became known as Shatranj, a name that survives in English words today. Most agree that chaturanga reached Europe by way of Persia, the Byzantine Empire, and the expanding Arab world. Islamic traders and travelers carried chess to North Africa, Sicily, and the Iberian peninsula, likely by the tenth century.
Along the way the pieces changed. The slow Indian elephant became a piece that could leap, and eventually the modern bishop. The queen rose from a weak piece into the most powerful on the board. The game absorbed ideas from every culture it passed through, which is exactly what makes its history so hard to write with confidence.
Medieval Europe turns it into chess
Europe adopted chess eagerly, and then reshaped it. By the late fifteenth century in European play, an enormous change was underway: the queen became the long-range powerhouse she is today, and pawns gained the option of moving two squares on their first move. With that two-square pawn move came the special capture known as en passant. The rules we play by today, including castling, stabilized over this period, commonly dated to the late fifteenth and sixteenth centuries in Spain and southern Europe.
Chess did not win favor everywhere overnight. It survived a series of prohibitions, notably from religious authorities who worried about its popularity and the gambling that sometimes surrounded it. Over time the game outgrew those objections.
The modern game
In the nineteenth century, organized chess took off. Tournaments became regular events, and in 1886 a match explicitly played "for the Championship of the World" produced the first generally recognized world champion, Wilhelm Steinitz. In 1924 the Fédération Internationale des Échecs, FIDE, was founded in Paris to govern the game internationally, and it still runs the World Championship today.
So the game you sit down to play carries a thousand-year story in it. The knight still jumps like a horse and the bishop still hugs its diagonal, echoes of the Indian and Persian worlds where this all began. What we call chess is the product of every culture that spent a century or two making it their own.