This would probably be The Royal Game of Ur AKA The Game of Twenty Squares, which is around 5000 years old. In 'Ancient Board Games in perspective', Irving Finkel writes of a cuneiform tablet dating back to 177-176 BC which:

"contains direct information about: 1 the gaming pieces, 2 the dice, 3 the throws needed to launch each piece, and 4 the effect of the individual pieces having either landed on or failed to land on the marked squares of the track. This tablet is thus unique among surviving cuneiform documents, and is by far the oldest attested example of rules for a board game."

Any Egyptologists here to provide more info on the game of Senet?

We don't have the exact rules recorded anywhere, but there are several mentions in Roman poetry of the game mechanics of the ludus latrunculorum, or the "Game of Robbers". The game appears to be similar in some way to draughts or checkers, with two players attempting to capture each other's pieces on a square grid, usually of 8x8 squares.

Our best description of the game itself comes from the Laus Pisonis, a praise poem from around the 2nd Century CE which details its subject's skill at the ludus. Here's my translation of the Latin:

Laus Pisonis 190-208

"If by chance it aids you, tired by the weight of your studies

Not to be weary still and to undertake gaming with skill,

With a more clever technique a piece is moved onto an open space

And wars are waged with a glassy soldier,

So that a snowy piece checks black ones, and now a black checks whites.

But who has not shown their backs to you? What stone yielded while you were its leader?

In a thousand ways your formation fights: this one, even as he flees,

Himself snatches his pursuer; that one goes on a long retreat,

Who once stood on the lookout; here one dares to commit himself to a brawl

And misleads the enemy, coming at his prey;

That one undergoes flanked delays and though he seems to be in a snare

He himself ensnares two others; this one is moved to greater things,

Driven into motion so that he can burst through the broken enemy columns,

And closed walls are laid to waste in a low valley.

Meanwhile, the most severe battles surge with cut out

Soldiers, yet you prevail with a full phalanx

Or robbed only of a few soldiers,

And the captured crowd clatters in each of your hands."

From this poem, as well as from other descriptions of the game in Ovid, Martial, and Seneca Maior, we can guess that the game pieces would move orthogonally across the board, much like the rook in Chess, and that an opponent's pieces were captured by flanking them with two of one's own pieces.

Beyond that, we don't know much about any nuances of the game or specific strategies, but we do know that it was incredibly popular: there are game boards etched into stairs and floors across the Roman empire, pieces and boards have been found in tombs, and off-hand references in authors like Varro and Seneca Maior indicate widespread knowledge of the very basics of the game's mechanics.

As a corollary, what is the oldest known game of any variety people played?