There are many great games designed specifically for two players, providing an enjoyable and competitive experience. Here are some popular options across different genres:

Board Games:

Chess: A classic strategy game that's easy to learn but difficult to master.
Checkers/Draughts: Another classic board game with simple rules but strategic depth.
Jaipur: A fast-paced card game set in the world of trade and markets.

Card Games:

Rummy: A matching-card game where players aim to form sets or runs of cards.
Lost Cities: A card game of expedition and exploration with a balance of strategy and luck.
Uno: A simple yet fun card game where players try to be the first to get rid of their cards.

Strategy Games:

Patchwork: A two-player board game where players compete to build the most aesthetically pleasing quilt.
Hive: An abstract strategy game where players maneuver insect-themed pieces to surround the opponent's queen bee.
7 Wonders Duel: A two-player version of the popular civilization-building game.

Cooperative Games:

Pandemic: Legacy Season 1 or 2: A cooperative board game where players work together to save the world from diseases.
Forbidden Island/Forbidden Desert: Cooperative games where players collaborate to complete a mission on a sinking island or a desert.

Card Drafting Games:

Splendor: A game of gem trading and development where players build their economic engine.
Sushi Go!: A fast-paced card drafting game with adorable sushi-themed cards.

Abstract Games:

Azul: A tile-laying game where players compete to create the most visually appealing mosaic.
Quarto: An abstract strategy game where players aim to create a line of four pieces with a common characteristic.

Deck-Building Games:

Dominion: A deck-building game where players strive to build the most efficient deck.
Star Realms: A space-themed deck-building game with a head-to-head format.

Quick and Light Games:

Love Letter: A deduction game with just 16 cards where players aim to deliver a love letter to the princess.
The Fox in the Forest: A trick-taking game designed specifically for two players.