If you have a dog then you ought really to have a first aid kit for them. When we go camping we put 2 first aid kits in the car - one for us and the other for our dog. As you know dogs can do some silly things sometimes and certainly our dog has no concept of danger if there is a cat to chase or something else interesting around. Unfortunately for us out dog suffers with grass pollen - I'll write about that later - so we have to have quite a few things for skin rashes and cleaning eyes.
Cuts and wounds
We had one occasion when she stood on some glass and so needed to stop the bleeding before we could get her to the vet. Therefore we always put some gauze and tape in the doggie first aid kit. Apparently hydrogen peroxide is very good for cleaning wounds but I haven't used that myself.
Dog in shock
If you have a dog that has gone into shock then as with humans you must keep them warm so keep a clean large towel or blanket in the car just in case.
Bites and stings
Always include some antihistamine in case your dog gets stung and you can also get antibiotic gels and eye washes. You can always prepare your own emergency eye wash using boiled water (cooled) and salt.
General items
Cotton, gauzes and cotton pads are good for mopping up and removing dirt. Having had a holiday in a location where there were lots of ticks we now always have some tweezers and TCP in the kit. Scissors (safer with rounded tips), syringes for giving medication by mouth and elastic bandages are also a good idea. I know some people keep a rectal thermometer in their kit but I don't personally - I'm not happy about doing that myself!
We also have a Labrador care book that we take with us. Its full of great information and advice with examples of first aid. Remember to have your vets number on your mobile or if you are in a different place look one up and put it in your phone temporarily.
Some dogs can be aggressive when injured or scared so depending on your dog it might be a good idea to put a muzzle in your kit. Having said all this I hope you don't have any occasion to use your dog first aid kits.