One of the easiest things to do is ask for stories from the group. Hint: Plan ahead, ask early, and require that the storyteller know the ending before starting, otherwise it could go on and on and on. If there are a lot of takers limit the time each participant can have. Making a contest out of it is generally not a good idea, especially where children are involved. Things can get out of hand quickly, and some of the shier participants may back out once the stories get going.
Contests:  If you want to have a contest, then plan one specifically for one night of multi-night camping. Make it an event, with snacks, and a specific format. A popular format that has been used successfully is to require two rounds of storytelling with a 'fire snack' break in the middle (i.e. Marshmallows, or Smores).  Some times it is hard for younger campers to tell stories, so allow jokes and 'How-To do something'.  Be prepared for silly, but the idea is to have fun.

A real group favorite: Start with one of the group and the Letter A, move around until you get to the letter Z, it goes as follows:
I went to the store and bought an apple.(NEXT)I went to the store and bought an apple and a banana. (Next) I went to the store and bought an apple, banana, and some crusty cottage cheese… And so forth.
Multiple word additions like 'Crusty Cottage Cheese' are real memory testers when you begin to get halfway through the alphabet. Works with any size group.

A real group Favorite: Start with one of the group. Give the first line of a story, and continue around the fire with each of the group taking his turn adding something original to the story.
There was a bear with a red hat. (NEXT)He wore it only on Sunday's and Wednesday's. (NEXT).

A real group Favorite: Quote recognition. Going around the fire, each of the group relates a quote, and the others try and guess it. Limited to movies or television seems to work best.

A real group Favorite: Freeform Joke telling. Everyone gets a turn, or two, or three.

A real group Favorite: Singing songs…But this is a storytelling site…so..

NOTE: Skits can be great, but they work better for large fires with an area specifically designated to perform the skits safely.