With the Angels, I feel I need to have a separate list for 2002, and another list for all the other years.
2002:
Darin Erstad make the last out to clinch the World Series title.
Scott Spiezio's home run in Game 6 (Game 7 doesn't happen without it)
Adam Kennedy hits his third homer of the game to put the Halos ahead to stay in the ALCS clincher
Non-2002:
Clinching the first division title and playoff berth in 1979
Vlad Guerrero's two-out single for the tying and winning runs in the 2009 ALDS to clinch the sweep against the Red Sox. Angels had a bad postseason history against Boston up until then, and it was nice to get one back, although they lost to the Yankees in the ALCS
Nolan Ryan's no-hitter against Detroit in 1973 (he had three other with the Angels, but I've read this one was the most dominant)
Honorable mention: I'll always remember the regular season game in the 1986 season when the Angels were down 12-5 against the Tigers going into the bottom of the 9th. They whittled the deficit down to 12-9, and the Tigers brought in reigning AL MVP and Cy Young winner Willie Hernandez to close it out. Bases full, shortstop deck Schofield up, and Hernandez quickly goes 0-2 on him, and Schofield sends the next pitch into the left-field bullpen to give the Angels an improbable 13-12 win. Result didn't mean much in the long run, but it was a heck of a moment.