This happened to us in June. We were on our way to Seattle and were at our layover in Dallas. They cancelled our connecting flight and said all the rest of their flights for the remainder of the day were pretty full. Everyone was told it was first come first served on getting to Seattle on any other flights with their airline or anyone elses.
Six hours later, after trying to go standby on three different flights at three different concourses, we were able to get on a flight to Seattle. It wasnt fun hauling three little kids and a carseat all over DFW for those six hours, but it beat the original arrangement the ticket agent was able to give us of waiting until 10:00 that night, then flying to Las Vegas, then flying to Seattle.
So basically, they will try and get you on another flight, but remember that everyone else on your flight also needs to get to that destination and there are only so many available seats.