Sword Art Online has gone through a few different iterations since its release in 2008. In fact, the game that you see now is actually a remaster of an older title. The best way to play SAO would be to start from the beginning and work your way up, but these are all great games to play as well.
The World of Swords: This is the original Sword Art Online game, published in 2009 and set before the events of the anime/manga series. It focuses on the battle between Kirito and Asuna against Fayeza Novus, leader of the Alicization movement.
Beater: Beater is an minigame that was added to The World of Swords as part of a free update in late 2018. You control someone called a «Battler» and have to battle through waves of enemies while avoiding deadly traps.
Red-Nosed Reindeer: Red-Nosed Reindeer is another DLC add