My dog attacked a puppy

That is a very sane response, BourbonSt.

I am late getting to this thread but I also think the way Boxer puppies play is a problem for some dogs. Timber was playing with a boxer puppy in my friend's living room when suddenly he ferociously growled and snapped at it. he never touched it but the puppy yipped at the top of his lungs and jumped up on the couch shaking like a leaf. I was stunned. To my knowledge an adult dog would normally just growl a warning to let a puppy know he had enough but Timber apparently felt he needed to send a stronger message. I have seen all different breeds gang up on a puppy at a dog park. it just takes one dog to single out it's prey so to speak and the rest feed off of that energy. I think the woman that owned the puppy shouldn't have brought it there in the first place. at the dog park here  they have a separate area for puppies and small dogs but this pupppy was in with the big dogs. the owner was just standing around talking to a group of people and only noticed what was going on when the puppy started screaming. luckily she was able to get it before it got hurt.  the dog that instigated the whole thing was a husky mix and the owner was watching the whole thing. I yelled at her to get her dog away from the puppy. nobody except me seemed to think it was any big deal.  Lizzy is used to playing with large dogs but I won't take her to a dogpark until she is an adult.

The fact that there are four children in the home makes me think the shelter should have placed a different dog with you if they knew yours to be a "terrier mix".  Kids like to take their dogs out and about, and now yours must be denied this privilege.  And, I submit that you should not be taking your dog to any dog park where there are small dogs or pups on which he could perpetrate the same mischief.  What must be considered here is that he didn't just grrr, or body slam, or lay over the pup, he put teeth marks in it sufficient to incur a vet bill, and you must never let that happen again.  Perhaps the answer is to contact the owners of dogs that he does get along with (preferably female) to arrange private play dates.  The owners need to know he damaged a puppy before they take the chance on having their dogs play with him.  Personally, if you brought him to the dog park and he injured one of my dogs after you already knew he had damaged another dog, I would be one hot mama and you wouldn't want to have to deal with me.
That said, I do think that your dog can be a good companion for humans.  Dog aggression and human aggression are usually two different things.
As to whether your command would have worked, it might have - that's why people will often tell you that if you think an off leash dog is going to attack you, try asking it to "sit", which most dogs know as their very first command.  Believe it or not, some dogs actually do it, and that gives the attackee an out.
Now that you have an idea that your dog is a Pit mix, be aware that you must take precautions around other dogs. 

ORIGINAL: spiritdogs

The fact that there are four children in the home makes me think the shelter should have placed a different dog with you if they knew yours to be a "terrier mix".

I think there is a lot of denial about these mixes. The shelter people have good hearts but they are in denial that there is anything different about a pit-mix. He was only 6 or 7 months old when they placed him with us and of course they never saw any animal aggression - he wasn't old enough. They are not thinking about the fact that the animal aggression is built in. You would think people who work so much with dogs would have known better, but they are also trying to place as many dogs as possible and I think they consider it a successful placement if the people walk out with the dog.

I have also realized, by reading so many of the threads here, that pit mixes are not the only dogs who might behave like this. I could have adopted a different dog and he could still turn out to be dog-aggressive. I've seen some pretty aggressive dogs at the dog parks, and they're usually not pit bulls. The day the puppy was attacked, the most unpleasant dog there (snarling, snapping at everyone) was a poodle! So you just never know.

Most puppies are just trying to find their way in the world.  They are probably giving off the wrong signals and your dog is taking them wrong.  If he's not drawing blood he's probably just edjucating the pups.  My Ridgeback got her ass handed to her when she was growing up by another bitch.  My girl was prancing around harassing everyone and very dominant but came up to some equally dominant but more mature dogs and they set her straight about dog park language.  I'm no expert but just my observation and thoughts.

ORIGINAL: BourbonSt

Thanks for those links. So apparently I can't bring my dog to the dog park anymore. I guess it would have happened sooner or later. Hopefully this little puppy will be ok. I have no idea how much damage was done to him and I'm terrified.

Anyway, my dog really does love playing with other dogs. Obviously the dog park is out. Does anyone have any suggestions on how to find a play group or something? Would it be wise to try to let him play with other dogs in a different setting?

Not if he's a Pit or Pit mix that has shown aggression to other dogs (and an adult that causes blood to flow on a puppy has shown aggression).  Some dogs should just not be in off leash situations with other dogs.  Among the breeds that this can happen to are Pits, Akitas, and others, but it doesn't make them bad dogs, it just makes them dogs that aren't ok at the dog park or loose with other dogs. 
Dogs that are just "disciplining" an obnoxious pup don't draw blood, so if yours did, and it was a bite, not an accidental tear, keep him out of play times.