A satellite state, also known as a puppet, Sub-state, dominion or vassal, is a country which allows another nation to represent it on the international stage. The satellite state remains independent in terms of government, economics, and other internal affairs, but may not sign treaties or declare war on foreign countries. The "mother country" retains special privileges, being able to sign defensive alliances and military access treaties with vassal state. If the mother country is at war then the satellite(s) or dominion(s) will automatically be at war with the enemy too. A nation can force another nation to be a puppet state by war.
However, great powers can still attempt to place the satellite state within their sphere of influence or, if the satellite's government allows it, invest in the local economy. The mother country (if a Great Power) receives 100% bonus influence with its satellites.
Formable countries can still be formed by influencing a puppet nation even if the forming nation is not the liege. This can be used to "steal" other nations puppets by forming countries. The liege can not do anything to prevent the formation. This is typical for NGF and Holstein. If one intends to form a country where ones puppet is needed, is it NOT enough to be the liege. The only way is putting it in ones sphere of influence.
Dominions are countries that are your same culture group and are your puppets. If you're Spain and you vassalize Portugal, Portugal becomes a dominion instead of a satellite. However if France vassalizes Portugal, they become a satellite. Dominions are more deeply bound to the liege than Satellites.
In HOD, one can take direct control over ones puppets troops.
If a satellite becomes a Great Power it automatically breaks free of the mother country. A succesful revolt in either the liege or the puppet nation will also make the country free.