HEADING FROM CANADA INTO THE U.S.
U.S. border crossing requires a valid rabies vaccination certificate for dogs and that both dogs and cats be free of disease communicable to humans. Note that pet foods containing lamb, sheep and goat products are not allowed into the U.S. (you may be asked to show labels). All the details are here.
HEADING FROM THE U.S. INTO CANADA
Generally, most dogs entering Canada "on holiday" with their owners must have a valid rabies certificate of immunization. The exact requirements vary depending on the age of the dog - see the government website for exact details.