A certificate of health is valid at a national level. However, if we want to take our dog outside our country, it needs to have a passport. It will have the information about its vaccines, the date in which they were administered, and the date in which they need to be applied again. It must be signed by the veterinary along with the id number of his/her college.
Both the information on the owner and the veterinary doctor are in the passport. If the dog has a new owner, there is a space to write this new information, as well as the information on its microchip.
There are some european countries in which only a passport is needed. There are others, like England, which need an anti-rabies certificate. It is always advisable for us to investigate about such matters before actually doing the trip, specially if this trip will happen in Europe.