I know that in the parks there definitely are, and each park is probably going to have its own rules regarding permits, reservations, durations, etc. Most of them will have designated campsites, even in the backcountry.
I'm not sure what the rules are if you're not in a state/national park. Did you have somewhere specific in mind?
Every campground will have a different set of rules regarding time limits and cost, but Colorado does a nice job of making the campgrounds very nice and easily accessible. They also make the information you need very visible when you pay your fees. If memory serves, most limit the time limit to about 14 days, but I haven't camped in Colorado in a while, so it may have changed.
I've been curious about this as well. Trying to get the wife and kid up there to do the Alpine Loop this summer, and was thinking how awesome it would be to skip the hotels and B&B's, and just camp along the trail.