Pros:
They come in a huge variety of sizes, shapes and colors. A breed for everyone.
They (homers) can fly home from hundreds of miles away, which is absolutely incredible.
Rollers!
They are addictive, as others said. I put this as a pro because they are so much fun.
Don't require much space.
They don't eat that much.
They can fly!
You can send messages with them.
Their coos are quiet and relaxing.
They breed readily on their own and are very prolific.
They are hardy and tolerant of almost all weather.
Very inexpensive.
They are easy to tame and make good pets.
You can also raise them for meat.
You can show them, race them, have rollers/tumblers, or just have them for fun.
An excellent chance to get to know your local birds of prey very well, as you can expect them to start hanging around a lot once they find out you have pigeons.
Pigeons are fabulous! There are few people I know who started with pigeons and didnt get completely addicted.
Cons:
Pigeons are a favorite food of many raptor species.
They are not particulary more messy than chickens. They poop more because they have a faster metabolism. To me, messiness is relative to the # of birds you are keeping in a given space and how well that space is maintained. But they do not spend much time on the ground so you cant just lay shavings down (If you do, they just get blown about by flapping wings). A regular scraping of perches/shelves is necessary.
Most people seem to hate pigeons. Expect looks of disgust when you tell people you have pigeons. Sorry, its just the cold hard truth.
Pigeon-keeping is a dying hobby.
Despite the cons, I highly recomend pigeons. I got my first pigeons 5 years ago and now I cant live without them!