Bear in mind that you are never required to volunteer to be bumped off of a flight. The best way to avoid being involuntarily moved to another flight is to prebook seats. It is important to note that not all airlines permit this, and even the ones that do have a threshold set where even reservation agents simply can't reserve a seat number for you in advance. The percentage varies, but most airlines that allow for seat reservations cut them off when the flight is between 65 and 85 percent sold.
If you are traveling on more than one airline, consult with each airline to see if you can prebook your seats. Even if it is an undesirable seat to you (a middle seat, seats that are rows apart when you are traveling with other people, etc.), at least it is a reservation and they can try to improve it at the airport. Many airlines charge for prebooking a seat, but it can be money well spent on a busy flight.
The other way to best avoid being involuntarily moved from your flight is to check in as early as possible. There are some who will say you can get the best seats if you check in at the last minute, that they will bump you up to business class. This may happen in rare instances, but it is not a rule of thumb! Checking in at the last minute for a full flight often means a middle seat in the back row which is far from amusing on a long flight.
The tone of this should not lead you to think that every flight is hugely oversold, and people are being moved from their flights all the time. Certainly oversells occur more frequently at holiday times, but they can occur if other flights have been canceled or if the aircraft has been downgraded (changing a plane from a larger aircraft to a smaller one).
Even if you do volunteer to take a later flight, you may still be taking your original flight because a flight may break open (seats become available because people haven't shown up) minutes to departure.
But if your flight is full, and you don't mind lounging around in an airport, or traveling the next day it can be quite profitable getting bumped!
One of my favorite examples of this is a day where I was the agent at the gate for flights to Miami during spring break. There was a family traveling, but they didn't need to be there until two days later. Well the flights were oversold the entire day, but there were four flights to Miami. They volunteered to be bumped, and did so for the first three flights, and took a flight out at 7pm after racking up $900 each in compensation. They were given lots of meal vouchers to pass the time munching away. It was a family of four and I don't know if I have ever seen passengers happier to miss flights than they were, but I am quite sure that they smiled all the way to Miami.
