Both dogs and cats are social species which means they are born with the capacity to learn specific social behaviors and the appropriate social responses of their species. But they are not born with all they need to know. Some behaviors have to be learned. They have to learn many of the basic social skills so it is very important for puppies and kittens to have socialization time with other animals. Puppies and kittens need to have pleasant experiences with others of its species early on and often.
Dogs use body language, including several play solicitations, to communicate their intentions. One of these play solicitations is to just drop to the ground. Another play solicitation is called a play bow, where the fore quarters go down and the hind quarters are sticking up in the air. Another is to raise one fore paw either while standing up or in the play bow, and wave it at the other dog. These are all forms of inviting the other dog to play.
Problems can arise when another dog doesn't recognize the meaning behind these play gestures. A dog that lacks significant social experiences may be clueless about what a play bow means and could respond with fear. They may even attack the dog that was offering the play solicitation. But if you have dogs that have grown up with other dogs and have learned dog etiquette, your dog will understand the request to play. Dog etiquette is when a dog understands that if another dog does something, I'm supposed to respond like this. Then when one dog does perform a play solicitation, the other dog would know to play back if they feel like playing. Or, maybe the dog doesn't feel like playing, in which case he will ignore the play bow or just lie down. Then the dog wanting to play will need to know that his request has been denied. So then, he has to learn when to back off and not force the issue.
There can be conflict in the home when there is a young dog who likes to play a lot and an older dog who is fine playing sometimes but not always. The younger dog may want to play longer and harder than the older dog. Sometimes the older dog will play and other times it will just ignore him. The younger dog has to learn to back off and give the older dog his space.
This is all part of the socialization skills that dogs need to learn so they can interact with other dogs. They will need these social skills when they go to parks, parties, gatherings and even out to local pet stores. Help your dog learn these skills early on and you will be rewarded in the long run.