'Vomit Comet' is a nickname, originally hung on any aircraft being used to provide simulated weightlessness for the purposes of training pilots and astronauts. In 1957, the USAF began operating flights of this type, and the subjects promptly hung the moniker on it because of the parabolic arc flight path ('comet') and the fact that, yes, it tended to make its passengers nauseous. In practice, a modern jet airliner or transport can, by careful management of altitude and speed, provide around 20-30 seconds of zero gee per parabolic arc without exceeding its safe flight regime. This is followed by a period of greater than 1G (generally 2-3G) as the aircraft pulls out of the dive the previous arc left it in and begins climbing into its next arc. This rapid change of experienced gravity, from zero to +2G or more, is what tends to cause nausea (on top of normal airsickness cues from riding in an aircraft while not securely belted down).
The name for the general type of aircraft used for this operation is 'reduced gravity aircraft.' Various nations and operators have used a wide variety of aircraft types to fill this role over the years. The initial USAF program used a Convair C-131 Samaritan. In the US, NASA took over the operation of these training flights in 1973, using converted ex-military aircraft (a pair of KC-135 Stratotankers) for the purpose. Other nations, of course, operate similar flights and programs; the nickname 'vomit comet' applies to them as well. The NASA KC-135s were used to film several spaceflight movies, most notably Apollo 13, to provide 'actual' weightless filming conditions.
The NASA aircraft retired in 2004 and 2014 as they aged. In 2008, operation of these flights for NASA was taken over by a private company. Zero Gravity Corporation currently operates a Boeing 727-200 outfitted for zero-g operations to provide both government-contracted training, and to provide private enthusiasts an (expensive) way to get the feeling of weightlessness. That airframe, despite being a relatively old design, was selected because of its very high G ratings providing an improved safety margin, permitting them to remain under 1/5 of the designed G loads during parabola operations (especially with the aircraft as very lightly loaded as it generally is) rather than staying at 1/2 of designed G loads or under as would be the case in most modern transports. The 727 design served as a freighter for a long time following its retirement from passenger service due to its durability, its ability to operate from short (and rough) landing strips, and a high parts availability due to the class's retirement. The current Vomit Comet in Zero-G service is a great way to experience flight on a 727, a rarity for passengers these days - and the zero-g experience is a hoot, as this noder can attest. If you are offered the chance to participate, I humbly recommend scopolamine patches; they are like an anti-motion sickness superpower when used correctly.