Dogs were the most common pet, generally small, fluffy dogs that were considered a sign of faithfulness. Gisants (tomb sculptures) from the late Middle Ages often show such a dog lying at the feet of their reclining lord or lady. Indeed, dogs were reputed not to leave the body of a dead master, willingly dying themselves from hunger and thirst rather than abandoning the dead. (See more here on dogs in the Middle Ages.)
Today people try to make pets out of very large or fierce dogs that were originally bred for hunting or guarding or herding (I have seen people trying to keep a boisterous English Sheepdog in a small apartment, without notable success), but medieval people knew a pet dog should be small and portable.
In the original story of Tristan and Isolde, Tristan gave his lover a small pet dog with a magic bell on its collar whose ringing tone would banish all sorrow. Isolde deliberately broke the bell off because she didn't want to be happy when Tristan was far away. (Okay, this adulterous couple had issues.)
Cats were less common as pets. They were considered cunning, and indeed medieval bestiaries said confidently that the word catus meant cunning in Greek. As on modern farms, there would have been semi-feral cats living in the barn, catching and eating the rodents that also lived there. Kittens were as fluffy and adorable then as they are now, but they would not be made lap pets the way small dogs would be.
Birds in cages were another occasional form of pets. Exotic birds (like parrots) were unknown in the Middle Ages, but magpies were reputed to be able to learn human speech. Pigeons and doves were not pets but they were still semi-domesticated, as dove-cots might be set up on an estate where the doves could nest, until they became dinner.