(The following article pertains to radios used for hang gliding in the USA. This information likely applies in other countries as well, but perhaps not all.)
Generally speaking, the radios used while hang gliding are in the 2-meter range. From an FCC perspective, there are three kinds of frequencies in this part of the spectrum:
1) Ham frequencies
2) Business band frequencies
3) Other frequencies
Ham frequencies are the most commonly used. In order to transmit on these frequencies legally, a Technician level ham radio operator's license is needed. There are pilots who use ham frequencies without the license, but there's really no excuse for this. A Technician's license requires only a multiple-choice test (Morse code proficiency is no longer required), and study guides are easy to find on the internet these days. For an exam, contact your local ham radio club, and they'll be happy to help you through the process. You'll wind up with a call sign, and be nice and official and legal. Note that there are ham operators out there who get annoyed if they hear people using the airwaves improperly, who sometimes report offenders to the FCC, who could confiscate your gear if they catch you. In addition to pilots, drivers should get ham licenses as well.
Business band frequencies are allocated to certain groups, including the USHPA. There are five such frequencies assigned to us: 151.625 MHz, 151.955 MHz, 151.505 MHz, 158.400 MHz, and 151.925 MHz. In order to use these frequencies legitimately, you need the authorizations (PA and VA) from the USHPA, though if the pilot has the authorizations, the ground crew can use them as well under the same umbrella (allowing you to have a non-pilot driver, for example). PA and VA require a simple test from the USHPA -- contact an Observer or Examiner about that. Note that although the USHPA has use of these frequencies, it's not exclusive use, and in some areas, there may be a lot of other chatter.
In addition, there are some frequencies in the vicinity of 2 meters that are assigned to other purposes by the FCC and are not available for use by pilots. However, in certain specific areas, some pilots have realized that these frequencies tend to be quiet, and have adopted them as the local standard. This is a violation of FCC rules, and is potentially asking for trouble. If you happen to be in one of these areas, and the local pilots do this, exercise appropriate wisdom and judgment. In particular, don't invite problems by using your ham call sign on a frequency that you aren't supposed to be on at all. (Also note that many of the local pilots may not have done their homework and might not even realize that they are on an illicit frequency.)
2-meter radios are easy to come by. If you get a ham license, you'll probably find yourself receiving some catalogs in the mail before you even get your FCC paperwork (licenses are public record, and radio equipment shops are happy to welcome you to the fold with some advertising). There are also brick-and-mortar radio shops around, and plenty of internet dealers. As of the end of 2010, a 2-meter handheld costs somewhere in the $120-$150 range. The most popular radios for quite some time have been the Yaesu VX150 and VX170, which have been replaced by the FT250 and FT270. Other manufacturers make similar products, and it's wise to ask what the pilots in your area use, in case you want to get one where there's local expertise in how to operate it. There are also some much cheaper Chinese-made radios available on ebay, although it's unclear whether it's technically even legal to own them in the USA. They are not well suited for flying, but if you want something for family and friends to be able to listen to the pilots, this is an affordable option. A serious retrieve vehicle is a candidate for an installed radio with a good antenna.
Modern ham radios have a wide variety of functions, and come with a thick user's manual. Most of this information is not needed for basic hang gliding use, and you can get by just fine if you can key in a frequency, lock the keypad, and maybe adjust the transmit power. The commonly used frequencies can be entered ahead of time as presets, so that you don't have to fiddle too much with the radio while getting ready to fly. Make sure you know how to operate the basic functions of the radio, because it can be frustrating to be standing around near launch trying to remember some obscure button combination to get your radio working when you shoudl be thinking about flying. Don't be afraid to make a "cheat sheet" of the important functions and keep it in a harness pocket.
2-meter ham radios can transmit only on legal ham frequencies, though some (including the Yaesu models) can listen on other frequencies. It also happens that some manufacturers designed their radios with the MARS/CAP[] program in mind. This is a program in which specifically authorized people modify their radios in order to be able to transmit on a wider range of frequencies, to provide assistance to the military. Such a modification typically involves something simple like the removal of a single resistor, and the details for some radios can be found on the internet. Doing this would allow the radio to transmit on the business band frequencies, as well as the "other" frequencies, but modifying a radio in this way or using such a radio is illegal, even if the frequencies used are otherwise legitimate (such as the USHPA business band frequencies). The right way to transmit on business band frequencies is to use a crystal-controlled radio that works only on the specific channels. Such radios are, however, significantly more expensive than 2-meter ham radios, and are rarely if ever seen in a flying context. The cheap Chinese radios can typically transmit on all the frequencies, which is why they are likely not legal for sale in the USA.
2-meter radios are usually sold with a "rubber duck" antenna, which is... better than nothing. A better antenna is the easiest way to improve the performance of a radio. Antennas are available from ham radio suppliers as well as some hang gliding vendors. It's also possible to make an effective 1/4-wave antenna from a piece of coaxial cable -- ask your local ham club or other pilots for advice on that. Antennas work best when oriented vertically, and the harness mains provide a convenient place for that. Decide where you want to carry the radio, use an appropriate length of shielded cable to get to the base of the harness main, and run the active portion of the antenna up it. Be careful to keep the antenna clear of any significant pieces of metal (such as the carabiner), as that can change the effective length of the antenna and reduce its effectiveness. With a proper antenna, a radio used while flying can work over a large distance even on low transmit power (which extends battery life).
There have been cases of people attaching a handheld radio to the control frame with ductape. Although that can work, it's far from the best plan. The usual setup is to have the radio in some pocket of the harness, with a headset in the helmet, and a PTT (push to talk) button somewhere that it can be reached. Depending on which pocket is chosen, this will likely mean that the radio buttons and display are not available to the pilot during flight. You pick a frequency, and if the radio has a keypad lock function you should use it so that if the buttons get pressed while the radio is in the pocket, it won't change settings. Even with the radio in a pocket, it should be leashed to the harness in some manner in case the pocket pops open. As one experienced pilot has put it, "It's really funny when a radio falls from the sky -- as long as it's not your radio". You might want to put some thought into the radio position if you are not using an antenna on the harness mains, since the antenna does emit electromagnetic radiation which you might not want directly against your body for health reasons.
Like radios and antennas, headsets are available from various sources, or if you are ambitious you can make your own. Headsets made for motorcycling are sometimes used, and there are also some hang glider pilots who make headsets specifically tailored for hang gliding. A single speaker in the helmet is adequate (some pilots like to have it in the ear that is furthest from the vario). The PTT talk button can be located on a shoulder, on the helmet, or on a wire that is run inside the clothing to the hand and is mounted on an index finger where it can be pressed with a thumb. The latter has the advantage that it can be used with both hands on the control bar, while towing or thermaling. (Some pilots feel that if you can't get a hand off the control bar, you should concentrate on what you're doing rather than chat on the radio.) VOX (voice activated transmit) is not suitable for flying applications, because wind noise will cause it to transmit too often. Some headsets also provide an external volume knob, so that the radio itself can be turned up to full volume, and the external knob used to adjust the volume in flight (useful if there is other traffic on the frequency, or when you just get sick of listening to the banal chatter of the other pilots). The wiring and connectors of the headset and PTT can be a constant source of problems, depending on the style and quality of the components used. If there is a part of the wiring that will be flexed often, the wires can fatigue and break. If there is any way that the connectors can be plugged in wrong (backwards, or the wrong connector in the wrong place), it's a good idea to put color-coded labels on them to reduce confusion. Connectors that don't have to be disconnected should be taped up so that they can't be, and some pilots prefer to install the wiring harness permanently in the harness. The only caveat is that you may want a way to remove the radio from your harness and attach a normal antenna so that it can be used as a handheld after landing.
Don't talk more than you need to on the radio. Leave the channel open in case somebody else has something important to say. Communication about where you're going, where there is lift, or advice to other pilots is fine, but discussions about whether your harness makes you look fat are a waste of bandwidth. In particular, once you land, it is appropriate to send a brief transmission indicating that you are on the ground safely, then keep quiet. (You should also move you glider to the edge of the field, or turn it tail into the wind, so that other pilots can tell by looking that you are okay.) Before launching, send a transmission saying "Radio check", and if you hear such a transmission, you can respond (e.g. "Full copy" if you hear it loud and clear).
Radios can be used for coordinating retrieve, but if there is cell phone coverage where you are, that may be a better choice. In particular, text messaging is the best way to communicate important information, for several reasons. One is that there is no risk that the communication will be cut off or garbled, which is particularly important if you may be in trouble. A text message will either go through correctly, or not at all. Another is that it can be easier and more reliable than something which needs to be written down. If you are sending your GPS coordinates so that someone can come get you, it's a lot better to send them as a text, so that the driver can key them in when he has a chance to pull over, than to have him trying to jot numbers down while he's driving.