Does each airline have its own set of dispatchers or do the dispatchers work for the airport?

Some great responses here, so I'll elaborate only just a little further.

Having worked as a 'dispatcher' for both a Part 135 air courier and at an airport (for a ground handling company), the previous entries make an important distinction: true dispatchers go through a course of training to become licensed / certificated under Part 121 and are normally employed by the airline. Housed in their Operational Control Centers (OCC) or System Operations Centers (SOC), dispatchers are likely on shift schedules, sometimes 24/7 as with my company. A former colleague recently got hired with United Airlines after tours with a couple of smaller carriers and is now enjoying a great salary.

There are 'dispatch' positions listed by ground handling companies but these are really operations agent positions: working at a local airport, you are the main point-of-contact for the air crews, ensuring such things as proper catering, weight and balance, flight paperwork, etc., and you often liaise with the different departments(ramp, pax service, the airport). This might be a great way to some basic training; most of these positions are hourly. A recent example was an Ops position I turned down with Southwest because it started at $9.50 an hour (I was making substantially more) but increases were regular due to the union contract.

It's still possible that I may go for my certificate, and I have a couple of training options here in FL; one school, Sheffied.com has a combo online and residence program, where you attend in-person at the end of the course and take your exams.

And one more important suggestion: if don't already have some flight training, go take an introductory flight - see more at http://flighttraining.aopa.org. It will help feed your passion for aviation, just as it did for me. All The Best to you in your career pursuits!

At present I believe only Delta & ExpressJet dispatch offices are at the airport proper in Atlanta, GA. Southwest's dispatch is on Love Field. American's dispatch is just of DFW. Continental Airlines dispatch office was in downtown Houston at 1600 Smith on the 13th floor. That's right the 13th floor. United Airlines was located out at ORD but during the UAL/CAL merger, the new UAL/CAL HQ was the Williams Tower in downtown Chicago on I believe the 25th floor. So the dispatch group of both CAL & UAL were both equally unhappy campers. Virgin-America's HQ/dispatch is at SFO while Alaska is just off the SEA airport proper. Understand that with modern communications that face to face briefings are no longer required as dispatch was in the days of old. When NWA & DAL merged, the NWA dispatch office in Tokyo Narita was closed. NWA dispatchers from the USA had bid by seniority to work in Japan. Nice huh? During Continental's second bankruptcy, the bankers required Continental to split off their Air Micronesia operation as that this was profitable. So a dispatch office was opened and staffed by locals from Guam, Saipan & the Philippines. Then just before the CAL/UAL merger, then Air-Mic was folded back into CAL. Those Air-Mic dispatchers were brought into the CAL dispatch at the bottom of the seniority list. During the mergers, since most of the offices have been union, the seniority lists have been put together one for one so the seniority has been merged with little disruption. At present most dispatchers want to work at DAL, SWA with a consolation of either UAL or AAL. Hiring on with Alaska is a good start. Staying at ExpressJet or Skywest for any length of time is not recommended! There are several other commuter operations where the dispatchers are being treated as clerks, not even as assistant dispatchers. One dispatcher is being saddled with the responsbility for several clerks. Bluntly, we have Frank Lorenzo acolytes running a commuter industry. If you end up at one of these commuters, don't stay long.

Normally, you would work for the airline. Having some aviation knowledge and or experience would help too. It would also depend on what type of dispatching you are doing. Before I became a helicopter pilot. I was a dispatcher in a local hospital for our helicopter and ambulances. I had learned aviation lingo from the old flight simulator on the computer. Being around that made me want to fly.
Learn aviation lingo, and that will be a big help. You will only work for one airline though.

A licensed FAA Dispatcher is usually directly employed by a 14 CFR PART 121 scheduled airline. (Think United, Delta, Express Jet, SkyWest, and SouthWest Airlines) You will get the same training and education on rules and regulations and weather as the airline pilot. However you will not obtain the physical skills of flying the plane. Each airline has its own dispatch center with its own company employees. There is competition between employers for quality employees in this field.

Most 14 CFR Part 135 (Commuter and on Demand) Charter companies employ "communication specialist" that act in a similar role without the same responsibilities of a licensed Dispatcher.
However the focus is on customer service and interface between the pilots and the clients.

Each airport may have its own dispatch center however this is usually focused on ground vehicles and not associated with airplanes.

Hope this helps. If you have any more questions please don't hesitate to ask.

To tag onto Mike's answer above, it has become common even for Part 135 operators to hire FAA Licensed dispatchers (mainly due to the level of training this entails) as a preference over individuals without the license. If you are truly interested in working as a dispatcher for an airline, it would help to get your FAA license through a school that offers this training. Some college programs will offer it as an elective, but there are also independent programs around the US. I have known a handful of people who worked their way up from other jobs with an airline and had the airline send them to receive their dispatch training, but most obtained the license on their own.

Due to the FAA certification process that airlines go through, you would be employed by one specific airline under their operating certificate and need to apply directly with the airline(s) you are interested in working for.

www.dispatcher.org is a great resource for other information. Good Luck!