Simple jousting rules for 5e

   Attacks. Each jouster wields a shield (+2 AC if proficient) and a lance (1d12, use Strength for attack and damage rolls). On each pass, each jouster rolls an attack with their lance.
   Hits. If a jouster is hit, they must make a brace check (see below) with a DC equal to the damage taken. On a failed check, they are unhorsed and lose the joust.
   Brace check. A brace check is a Strength (Animal Handling) check.
   Initiative. Attacks happen simultaneously, and it is possible for two jousters to unhorse each other.
   Rules of the match. A jousting lance is destroyed when it hits. New lances are handed out and passes are repeated until one or both participants are unhorsed. Additionally, a limit may be set for lances broken; for example, the first jouster to break three lances (score three hits) wins if neither has been unhorsed.
   Passes. On each pass, a jouster may attack normally, or choose one of the following maneuvers:

• Aggressive: +5 to attack, -5 to AC. 

• Defensive: -5 to attack, +5 to AC.

• Braced: -5 to attack, +5 to Animal Handling.

• High in Saddle: +5 to attack, -5 to Animal Handling.

• Eyes Fixed: +5 to attack. If you are hit, roll a d20. On a 1, you lose your left eye if you have it, and on a 2, you lose your right eye if you have it. On a 3-4, you gain a Horrible Scar, and on a 5-10 you gain a Minor Scar. (See the DMG p. 272 for the effects of these injuries.)

The same information in table format:

Maneuver	Attack	AC	Animal Handling	Special
None	-	-	-	-
Aggressive	+5	-5	-	-
Defensive	-5	+5	-	-
Braced	-5	-	+5	-
High in Saddle	+5	-	-5	-
Eyes Fixed	+5	-	-	See above

King Henri II of France was mortally wounded by a splintered lance in the eye from the hand of Gabriel Montgomery, the captain of his own guard, in a jousting match in 1559.

Optional rules

Reading opponents. When a PC is jousting against an NPC opponent, if you wish, you may allow the PC to attempt to make a read. First decide the NPC's maneuver for the pass, and then allow the player to make a Wisdom (Insight) check contested by the opponent's Wisdom (Animal Handling) check. If the player wins the contest, they learn which maneuver the opponent has chosen, and can then choose their own maneuver based on this information.