As everybody has already mentioned, small pets can travel in the cabin with you.
And also as mentioned, there are many restrictions.
1) they have to be small enough to fit in a pet-approved carryon bag under your seat.
2) you have to pay a fee - varies by airlines. You cannot just show up at the gate with a dog/cat in your carryon. It has to be declared at time of ticketing. This is how airlines control the amount of animals on a flight AND how they can help allergic passengers avoid flights with pets.
3) pet is never allowed out of the bag - even to sit on the owner's lap. If you see a pet wandering the aisles, it is in violation of the rules and you should complain to a flight attendent. People who do that ruin it for people whose business it is to get dogs from one end of the country to another in short periods of time (show people).
4) There are only a few slots per plane for carryon pets. It varies between airlines, but it is very low, ie: 3 per cabin allowed.
5) This is the most important for allergy sufferers:
If you are severly allergic (ie: life threatening) to dogs and/or cats, let the reservationist know at time of booking. People usually come first and there will not be any pets on that plane. When you are travelling with an animal, you are under the understanding that if there is a life-threatening allergy on the plane, you could be bumped from the flight.
As for the comment that dogs belong in the cargo hold - I disagree.
Many of the people transporting dogs are going back and forth to shows. Many of these animals are worth sums in the 5 digits. Dogs die in the cargo holds. They get seriously stressed out. And if your job and livelyhood is showing this dog, you don't want it stressed out the day of the show. It is just as much a necessity for a person's livelyhood as an engineer's expensive laptop - only worth much more.
Airlines have regulations in place to protect seriously allergic passengers. The allergic passenger just needs to notify the airline and they will either bump the passenger with the animal or make sure the allergic passenger does not book on a flight with a dog.