I've phased the word "pet" out of my vocabulary for a few reasons. Fundamentally, I want animals to be seen in a more respectful way. Using the term "pet" can connotate a lower status, and a lack of respect. Does this mean animals are the same as humans? No, of course they have different needs and desires.
I prefer the term "companion animal" to describe dogs, cats, rabbits, mice, and any animal that is domesticated and has historically lived with humans. At this point, they are no longer wild and would probably not survive on their own. They have been bred to rely on humans for food, shelter, care, and warmth. We have an obligation to care for the animals that have been domesticated, and an obligation to work on cat and dog overpopulation by spaying and neutering our companion animals.
Many people who use the term "pet" often have a strong bond and love for their companion animal. They care for them as thoughtfully and with as much love as they would a human child. However there are others who use the term and have a "just a pet" mentality, and ascribe lesser worth to the animal. Using the term companion animal helps show the level of respect that you feel and animals deserve.
In addition, the term "owner" is disrespectful to our companion animals. Legally, it still be their status, but it implies animals are a piece of property, not a live being. When animals are viewed as property, people use this as an excuse for abuse and mistreatment. Using the term "guardian" in place of "owner" implies a more respectful relationship. It also will hopefully help people take a more vested interest in continuing to care for their companion animal versus abandoning him or her at a shelter.
For more information on this topic, please see In Defense of Animals' Guardian Campaign.