When humans travel internationally, they get a passport issued, so it makes sense that pets would need one too. However, there is a lot of confusion about pet passports, particularly in the US.
The pet passport concept was started for moving pets to the UK to simplify the process of importing pets from other countries with history of rabies. The passport itself comes in multiple forms, sometimes a pink piece of paper and sometimes a small blue booklet. It contains the microchip number of the animal, the certification that it has had a rabies vaccination, and needs to be signed by an officially approved veterinarian. The pet passport is technically only valid for pet travel to the EU and the UK -- however some countries will accept the vaccination records in the passport as the original vaccination documentation.
There are also several unofficial versions of pet passports available for sale on the internet and in pet stores. A good example is this Pocket Reference Journal found here. It's important to note that these are not acceptable for international travel. They are merely made to be an unofficial record of your pet's vaccination history and to be a convenient place to store everything.
Bottom line: If you're moving pets to the UK or EU, a pet passport can be a tool in moving your pets. However, having all of your original documentation can replace the pet passport -- in other words you don't have to have a little blue book. For all other countries, having your pet's original documentation (rabies certificates, vaccination records, blood tests, etc.) will be sufficient.