Newtypes (ニュータイプ) are people with heightened powers of precognition and exhibit various esper-like powers. Since they don't look physically different than any other person, Newtypes are considered to be the next evolutionary step for humanity living in space.
Sometimes people who aren't "awakened" as Newtypes are called the derogatory term, Oldtypes (オールドタイプ). Artificially engineered Newtypes are named Cyber-Newtypes (人工ニュータイプ, Jinko Newtype) or Artificial Newtypes (強化人間, Kyouka Ningen).
Background Info
The definition for this term has been very vague, but each character who is a Newtype can form various psychic links with other people. To what extent is different with each individual. Strong newtypes, like Amuro Ray or Kamille Bidan, can telepathically communicate to people who aren't even Newtypes. Usually, they can connect with one another on an almost instinctive level. If a particular Newtype is experiencing an extreme emotion, they can alert other Newtypes to their presence and may unleash powerful attacks. A few individuals can also seem to transcend death as they may continue to talk to others as a vivid spirit.
As a whole, Newtypes can normally sense danger before it happens. They can either detect it minutes in advance or a few seconds before it happens. When facing other Newtypes, they may feel a mental tension, which they usually call "pressure", from their opponents. Their powers can also cause painful side effects, especially for the younger and immature individuals.
While they are regarded as supernatural fighters and a useful military asset, their abilities make them loosely trusted by humanity. Char Aznable later uses the faulty acceptance of Newtypes to inflame the second Zeon rebellion against earth.
Mission Mode
In Dynasty Warriors: Gundam 2, the player can choose to become a Cyber-Newtype with any unawakened character. After they finish their characters' story missions, they'll be contacted by Nanai. She has an interest in the player's character and will offer them a test to complete. In reality, it's an experiment to see if the pilot is a worthy Cyber-Newtype. If they complete her request, they'll evolve into a Cyber-Newtype when they accept her second offer.
Alternatively, the player may wait until they are level 30 for Amuro to approach them and make a request. He suspects that their amazing pilot abilities are close to a Newtype's abilities and wants to test them. After a long battle in space, he congratulates them through the Terminal and realizes that they may someday surpass him. He warns them that their powers should only be used for the good of humanity before wishing them luck on the field. The character will become a Newtype afterwards. Naturally evolving into a Newtype is one of the requirements needed to unlock Quess and Gyunei and Amuro's Zeta version.
As the player levels up their non-Newtype characters in Dynasty Warriors: Gundam 3, they may naturally gain the Newtype status. Most case scenarios has this occur when a character reaches their maxed out stats or maxed out levels.
Becoming a Newtype or a Cyber-Newtype allows a pilot to use any Newtype related Mobile Suit, which are usually the ones with funnel capabilities.