Experience (or XP) is a form of currency obtained by defeating monsters in the dungeon, or through the use of special cards such as the Trainer in the Village. XP is shown in the lower left corner of pertinent cards, in red on monsters to show gained XP and in blue on heroes for the amount needed to level up.
Some players prefer using other items such as glass beads to represent XP, rather than the cards that come with the game.
If for some reason the XP pile runs out, player's continue to keep track of further XP gained through whatever means they can agree on.
XP is spent during a turn in the Village to level up a hero revealed by the player. This involves placing the XP cards or tokens back into the stock of XP for all players, destroying the hero card, and taking the next level version of that hero from the Village and placing it into the player's discard pile.
XP may also play other roles in later expansions.