Earnestness is a quality of sincere interest, belief, or opinion. The earnestness with which you recite your favorite poem might make your English class stop whispering and really listen.
Your favorite history teacher's earnestness can be seen in the serious, intense way he talks about the Civil War or Prohibition. Little kids often approach the world with an earnestness that's reflected in their amazement and wonder at seemingly ordinary things, like fireflies or simple magic tricks. Earnestness comes from the adjective earnest, "showing sincere conviction," which has an Old English root, eornoste, "zealous."