An occurrence, action, event, or happening that surpasses "epic" proportions. In order for something to truly be anthemic it must be worthy of songs of praise and glory written in its honor.
Wow, that final over time goal was anthemic!
The term anthemic is not strictly a word, yet is used to describe music that has a particular presence to it/atmospheric feel.
In hardcore/punk an anthemic band would be one who have plentiful singalongs and a tight, melodic, and often heavy rhythm section. Bands such as Bane and With Honor are a good example of this.
Statement: "Woah this band are amazing, every song is really anthemic"
Reply: "Yeah, they're incredible."