Artificial gravity

Artificial gravity is a technology employed by the UNSC, Covenant and the Forerunner, to generate gravity of varying strengths in an otherwise zero or low-gravity environment.

Centrifugal force
Prior to contact with the advanced technology of the Covenant, a number of UNSC ships employed rotating sections of the hull to provide centrifugal force on the interior of the hull to allow comfortable movement in "gravity." Carriers were able to adjust the rate of rotation, allowing more or less gravity, and a number of other ship classes, such as Halcyon-class Cruiser's, also used rotating sections before the introduction of reverse engineered gravity plating. The rate of spin for many of the smaller UNSC ships would have to be quite low (2 RPMs or lower) to prevent the crew from getting dizzy and nauseated due to the Coriolis effect. This means that the "gravity" produced by the rotating sections of most UNSC ships would be well below 1g. Space stations and colonies at least a mile in diameter would likely be able to produce 1g. But this style of artificial gravity fails to explain how an elevator can run through it and how you traverse out of the rotating section.

The Halo Installations rotate, and may use centrifugal force to provide at least part of their Earth-like gravity, possibly employing the same gravity generation techniques used by the Covenant, or another method, to generate the rest.

Linear Acceleration
A ship accelerating at 1G would be able to simulate gravity by using inertia. If certain decks within a ship were oriented to where "up" was the bow and "down" was the stern, then constant acceleration (or deceleration) at 1G would simulate gravity. Many UNSC ships probably used a combination of centrifugal force and linear acceleration to provide simulated gravity prior to the development/reverse engineering of Covenant technology. It's likely that space stations and some ships would continue to use these technologies, as they would not require an energy source beyond the ship's own acceleration or rotation.

Covenant Artificial Gravity
In contrast, almost all Covenant vessels employ some sort of artificial gravity mechanism that does not involve rotating hull sections or acceleration. Instead, their method allows for the generation of actual gravity. It has been speculated that they may use diamagnetism, but this is very unlikely: though there would be no known adverse health effects, it would mean that the entire ship would have to be made of non magnetic materials, have heavily shielded electronic equipment and the crew would not be able to use anything made of metal. The use of ultra-dense materials which generate their own gravity has been suggested, but this would make the ship incredibly heavy and massive. Considering the UNSC still use some form of reaction/rocket propulsion, it seems rather unlikely that this would be the case.

The most likely explanation is that the Covenant have developed some form of directional gravity emitter that generates a beam of gravitons. This would mean the Covenant are able to generate gravity without mass. It is possible that the Covenant have figured out a way to convert energy into graviton particles and direct them in much the same way as a beam of light; this seems to be the basis for the Covenant's gravity lift. The production of "antigravitons" would create a repulsive effect, which may be the basis for some of their antigravity drives. All this suggests the Covenant are able to produce gravity and antigravity as easily as humans can produce positive and negative electric current.

Although the UNSC experimented with true artificial gravity and antigravity prior to First Contact, most attempts were unsuccessful, or produced mixed results. They generated an artificial gravity/antigravity field, but they must not have been practical for unknown reasons. At least some passed into Insurrectionist possession. After contact with the Covenant, however, reverse-engineering Covenant technology allowed the UNSC to employ artificial gravity on most of its ships.

Discrepancies
It is widely agreed that the UNSC gained their current artificial gravity technology from the Covenant, and had to use rotating centrifuges to mimic gravity prior to this. However, the Halcyon-class cruisers and Phoenix-class colony ships clearly lack rotating centrifuges (although the former is described to the contrary in the novels), and both were originally commissioned far before first contact with the Covenant. Of course, it is possible that both classes were designed to lack gravity altogether, and were retrofitted with artificial gravity later in their service lives. However, if this is not the case, then it appears that the UNSC did in fact possess true artificial gravity technology before the Human-Covenant War.

The situation is further complicated by the human DCS freighters. Descriptions in Halo: Contact Harvest (set at the time of Covenant first contact) clearly suggest that some form of gravity was in play aboard the vessels, but the novel fails to mention its nature. It may be that antigravity plates (as seen in Halo: Ghosts of Onyx, said to have not "panned out") were undergoing trial runs aboard all of the vessels formerly mentioned, but this seems slightly far-fetched. Since the Antigrav plate had a crushing or repulsive effect, and it was described as generating a beam, it most likely generated directed antigravitons. If the UNSC was capable of generating true antigravity, then it is logical to assume they also had gravity generators.

UNSC starships clearly have a form of Antigravity, demonstrated by their ability to hover in planetary atmosphere without rocket propulsion being directed below. It's possible that only very large power plants can generate the power needed to produce antigravity, and the antigravity plate may be an early attempt to miniaturize the system found on starships. This would mean the human and Covenant technology were not that far apart in this regard, with humans lacking the technology needed to miniaturize gravity/antigravity generators. This would also explain why the UNSC was able to so rapidly reverse engineer and implement artificial gravity systems on board their warships: they may have been on the verge of doing so already.

Trivia

 * Eugene Podkletnov has claimed that he has designed a device that can create artificial gravity. In contrast, a group funded by the ESA claim to have developed a device that reduces the effect of gravity. The validity of these claims are unclear.
 * Artificial Gravity is a popular concept among science fiction novels, television shows and films, but attempts to explain their workings are usually implausible, involving gravity field generators.