3rd Strike tells the story of what happened after the events of Street Fighter III: 2nd Impact and the conclusion of the Third World Warrior Tournament (fourth, retroactively). Among the battles that occured during this time:

Alex vs. Ryu - Ryu encounters and defeats Alex comprehensively, leading Alex to become obsessed with fighting Ryu again and becoming a better fighter.
Ryu vs. Ken - The two friends face off once more and Ryu again narrowly defeats Ken, lauding him as his worthiest opponent.
Chun-Li vs. Urien - Chun-Li demands the return of a kidnapped child and Urien agrees to comply if Chun-Li can provide an entertaining match. The victor is unknown, although it is implied Chun-Li won, as the child is returned.
Necro vs. Twelve - Upon learning the Necro and his female companion, Effie, survived the events of 2nd Impact, Necro's successor, Twelve, is ordered by either Gill or Urien to hunt down and exterminate them. While the official outcome of this battle is unknown, Necro's ending reveals that he and Effie likely survived the encounter. Twelve, in the meantime, is returned to Dr. Woo's lab to recuperate, and it is hinted that he has gained sentience, as his hollow eyes gain red irises.
Akuma vs. Oro - Both have fought each other after an encounter, but declared to cease combat as there was no clear winner and both believed that a mutual death would occur had they continued.