Dog carriers come in great variety of styles now days , but not all are airline standard sizes.
The tag of the carrier will show you if it is airline approved. The carrier will have to be put under the seat in front of you just like a regular bag so make sure it will fit with your dog comfortably inside. You are only allowed one carrier per person. However, you can put two dogs in one carrier, as long as they both fit.
There are carriers which you can carry and also there are carriers on the wheels, which are also airline approved.
The official airline rules state that the pet has to stay in the carrier at all times. I have seen extremely strict flight attendants who have insisted that my puppies stay in his carrier on the floor at all times. Then, there are more flexible flight attendants who will allow you brief moment to take your dog out -if the carrier for some comforting
Some airports allow you to walk your dog on a leash through the airport. It is a great way for your puppy to stretch between flights. Just make sure there are no bathroom breaks taken on the sly. I usually carry a couple of puppy training pads in my dog carrier, so that way I can either use them at the airport, or in case emergency arises, then change the pad in the carrier. Also, good to have wipes, tissues and baggie, just in case.
My experiences, in general, flying with small dogs have been great. Once you learn all the details and tricks to make the flight easier, it will become a very positive experience.