It can depend on the type of ticket you purchased and what airline. For the most part, you'll be put on a following flight, depending on available seating. There might even be a small charge, some sort of processing fee, but I think that will not be likely in today's market. Airlines like to overbook a flight because so many people cancel or just don't show up (at least they used to back when business was good). This means that the next flight might not have an available seat and you'll just have to fly "stand by". That is, you sit and wait until they call your name about 5 minutes before they close the door (unless they're underloaded).
Good luck!