Trophies have been a Super Smash Bros. staple since Melee in 2001. Trophies are models of characters and things from various classic and recent video games. Each trophy comes with an often-interesting or amusing description, with a little insight into Nintendo and video game history.
You can look at your trophies in the Vault menu. You can buy them with coins you've earned from playing, or play coins. You can also simply find them in every gameplay mode. The fastest way to earn trophies is to play the Trophy Rush minigame. See the Trophy Rush section for more details about that.
Each character, including each type of Mii, has an "(Alt.)" trophy that only appears in the Trophy Shop, so be sure to buy any you don't have whenever you see them.