Parrots (including small ones likes budgies and cockatiels) are often more intelligent than cats and dogs, and many species live much, much longer. Macaws, for example, can live into their 80s (but 50 seems to be about the average). Parrots also form strong bonds with their humans and have plenty of personality. They are animals that NEED interaction, love, and attention. In fact, my cockatiel wants more cuddle time from me than my cat does. He likes to sit on my shoulder or snuggle under my chin for a few hours every evening while I watch TV or goof off on the computer, and he demands that I hang out in his room with him for much of the day if I'm home.
That being said, I think birds aren't as popular as dogs and cats because they have a reputation for being noisy, and quite a few people who like birds don't like the idea of keeping one in a cage (though people who properly care for their parrots give them lots of time out of the cage). They can also make a mess, but after working with dogs at an animal shelter over the summer, I have to say that birds aren't nearly as messy as many dogs.