Eagerness is a characteristic of being excited and prepared to do something. Shooting your hand in the air and shouting, "Me! Me!" when your teacher asks for a volunteer is a good way to show your eagerness.
A basketball team displays its eagerness by running onto the court at the start of the game, and a puppy's eagerness to lick your face and nibble your fingers is either the cutest thing about it, or one of the many reasons you don't like puppies. Eagerness and eager are both positive words, but in the thirteenth century eager meant "fierce or angry," and it shares a Latin root with the sour word acrid, or "bitter."