Article link: http://www.gamesradar.com/f/the-greatest-videogame-moments-of-2010/a-2010122310738775004
It doesn't have to be restricted to scripted events that happen in games but also things that you may have pulled off in a multiplayer match, or surpassed a gamerscore benchmark.
Without further ado, here some of mine.
1) The third act of Mass Effect 2, where you and your crew embark on the suicide mission. Everything about this part of the game defines epic and can be absolutely heartwhenching depending on the decisions you take throughout the game. I won't go into this further as to not spoile anything. If you haven't played this game yet, you own to yourself as a gamer to do so.
2) Wandering the West in Red Dead Redemption. Usually open world games lack in the visuals department to the vast nature of the game, (Fallout *cough*) but Rockstar made the downtime that MMO players know all too well into something a player would want to take time in doing. The landscape in RDR is one of the most beautiful environments on all of gaming. Just exploring into vastness of a snowy forests and getting ambushed by 3 cougars, only to get mauled by a bear while trying to escape is priceless, and something you can't experience anywhere else.
3) Arming all 6 six COMM sites in a round of Rush in Battlefield Bad Company 2 Vietnam. No words can describe the epicness.
Thats all I have for now. Feel free to add below.