That's so great that you have access to one in your community! So many places don't have dog runs yet:( I live in Manhattan, where there are about 31 of them. I've got one just 2 blocks away, and two others with in a 12 block radius! It's definitely one advantage of being an NYC dog owner. Here are the things that I see upsetting people and/or dogs...and for some reason, it doesn't seem to be "common sense" to everyone.
If a toy or object starts to create a problem, get rid of it. Playing with special things from home is a priveledge, not a right at the park. Some dogs make be too possessive to watch your dog play with a brand new frisbee while all they get is a dingy old tennis ball. Also, if you bring a toy, you must accept that it becomes community property until you leave (at which point, of course, you have every right to take it home with you). But while at the park, dogs may share or steal toys. As long as there are no fights, then it's fine.
Bring food and children at the run at your own risk, and knowing that it's your job to guard/watch them. You can't expect someone to keep yelling at their big dog (who may not be accustomed to small kids running around) to keep a distance. Also, dogs are going to come and sniff/beg your food, don't blame other owners for not keeping their dogs away.
Only run around if you sense that all the dogs at the park don't mind. But many dogs chase running people (my boxer sure does...and he tackles!) So if this happens, don't run around. It's exercise time for the dogs, not the owners.
Most other rules are a given...like breaking up fights, cleaning up after your dog, etc. It can be frustrating sometimes because people all have different beliefs on what "healthy playing" is. Some people truly belief that any mouthing is fighting and will ask you to break it up. And we all know Boxers like to make noise when they play, so people may tell you to take your "Pitt Bull" out of the park! Lots of people think mounting is totally inappropriate, but I tend to think that owners object to it more often than the dogs do.
Anyway, generally it is such a great time and you learn to ignore/avoid the neighborhood "crazies." And Michfont is right...you'll get addicted!