Universal Century Newtypes
In the Universal Century universe, Newtypes represent the next step in human evolution. They have a superhuman level of spatial awareness, and can sense the emotions of others (which is regularly used to detect and evade enemy attacks). Coupled with technology such as the Biosensor and Psycommu System, they perform seemingly impossible feats on the battlefield.
A Newtype is said to be able to utilize the entirety of their brainpower (as opposed to just half). This phenomenon most commonly occurs among "spacenoids," so it is believed that one must spend a considerable amount of time in outer space to become a Newtype. Newtypes were discovered during the One Year War, with several soldiers from the Principality of Zeon and the (in)famous Amuro Ray from the Earth Federation exhibiting Newtype traits.
Cyber-Newtype (or "Enhanced Human" in Japanese releases) refers to normal humans that are modified (through means unexplained throughout the franchise) to become Newtypes. These modified soldiers can then take advantage of Newtype-exclusive technologies. Early Cyber-Newtypes like Four Murasame suffer severe mental issues, but later ones like Gyunei Guss do not.
After the discovery of the Newtype phenomenon, several weapons are created to take advantage of their spatial awareness. Bits, Funnels, and Fin Funnels are wireless remote weapons that are controlled by the pilot's mind through the use of a Psycommu System. A wired equivalent for normal pilots, the Incom, is created, but is unwieldy and better used by a Newtype anyway.
A special Psycommu system, the Psycoframe, is used by Amuro's Nu Gundam to single-handedly push an entire asteroid out of Earth's atmosphere.
The Biosensor allows for particularly spectacular feats; it's used by Kamille Bidan to channel the spirits of the dead to create a gigantic beam saber, Paptimus Scirocco to cripple Kamille's mind, and Judau Ashta to move ZZ Gundam with mere thought.
After War Newtypes
Newtypes also appeared in Gundam X, and like in UC, have increased spatial awareness; they can also control pilot-less Bit Mobile Suits. However, unlike UC, Newtypes are described as mutants rather than the next step in human evolution.
Cosmic Era Newtypes
Although the term itself does not appear during Gundam SEED or SEED Destiny, several characters are described in secondary material as Newtypes, most notably Mu La Flaga. In CE, Newtypes have an increased level of spatial awareness, but are otherwise much weaker than UC or AW Newtypes. In the first SEED, Newtypes are the only ones that can control Gunbarrel- or DRAGOON-equipped mobile suits, but by SEED Destiny quantum technology is created to remove the reliance on heightened spatial awareness.
Other Gundam series
Although the other Gundam timelines don't feature Newtypes, most of them feature story elements that allows certain characters to either "sense" their opponent or gain precognitive abilities. These elements include Gundam Wing's "ZERO System," Gundam 00's "Quantum Brainwaves," and Gundam AGE's "X-Rounders."
In Video Games
In Super Robot Wars, "Newtype" is a special ability with up to nine levels that grants Accuracy, Evasion, and weapon range bonuses. Bit and Funnel attacks, as well as some other attacks such as Zeta Gundam's Waverider Crash, require a certain Newtype level. After War Newtypes are classified as "Newtype (X)," but otherwise receive the same benefits as their Universal Century equivalents. In Super Robot Wars A, Axel Almer receives the Newtype skill, though Super Robot Wars A Portable removes it.
In Artdink's Gundam Battle series, the player chooses whether their custom pilot is "normal" or a Newtype. Newtypes start with lower pilot stats but have a higher growth rate. Additionally, they can pilot suits with Newtype weapons, and the warning siren for an incoming attack is replaced by the famous "Newtype flash" sound.
SD Gundam G Generation includes an Awakening stat which governs specific weapons, including Newtype-exclusive weapons like Bits and Funnels. Many "Oldtype" characters either keep an Awakening rating of 0 or only receive a rating after obtaining a certain level. When pilot abilities were introduced to the series, Newtype became an ability that raises the Awakening stat.