Cheerfulness is what you feel when you're happy and carefree. Your cheerfulness first thing in the morning might be very annoying to your grumpy older brother.
Cheerfulness is a quality of brightness and optimism, a state that other people can usually sense from your happy whistle or the smile on your face. You could also use words like "cheer" or "happiness" to describe this sunny quality. This noun comes from cheer, which stems from a root meaning "face" — the idea is that cheer (or cheerfulness) can be seen in a person's facial expressions.