The Adams (アダムス, Adamusu) are four radiant giants present at at Second Impact which play an important ongoing role in the New Theatrical Edition continuity. Very little concrete information is available about them at this time.
The Adams as seen in their "giant of light" form are all slightly different in appearance, but all possess halos and wings. Fifteen years after Second Impact, the Gates of Guf are still open over Antarctica, with the four crosses presumably generated by the individual Adams still present. It is unknown whether the South Pole still remains in this state as of Evangelion 3.0.
Contrary to what the name implies -- i.e., that they replace the role of Adam from Neon Genesis Evangelion, and are thus the First Angel of this retelling -- they do not appear to be the First Angel collectively. In Eva 3.0, Kaworu Nagisa states that he himself is the First Angel (though he undergoes a "Fall" to the 13th as a result of the film's events). The lack of information about Kaworu himself, however, still leaves plenty of doors open to explore what precisely he meant by this claim, and what it means in terms of his true nature and connection to the other Angels.
However, it is known that Eva-13 (dual-piloted by Kaworu and Shinji) is an "Adams survivor" (Adams no ikinokori). This is revealed by Mari when Eva-13 becomes a radiant giant and transcends DMS (a.k.a. Eva Pseudo-Evolution) Phase 3. Because the Adams were themselves radiant giants, and "pseudo-evolution" can be taken to mean "not evolving into a new form, but returning to an old one", the meaning of "Adams survivor" becomes clear: Evangelion 13 was one of the four giants involved in Second Impact, and survived the event only to be captured by humanity and modified into an Evangelion.
Eva-13 bears a striking resemblance to Eva-01. Furthermore, the luminous being that Eva-01 pseudo-evolves into at the end of Eva 2.0 at Near Third Impact (a "continuation of Second Impact") resembles the "three-eyed" Adams seen at Second Impact. As a point of interest, the three-eyed Adams has its chest area cropped out of frame, preventing the viewer from confirming whether or not the two cores that Eva-01's radiant giant form manifests were originally present on that specific Adams. However, a three-eyed being, visually recalling Adam's giant of light form from NGE, appears on the IPEA's logo; and if the logo's design is interpreted a specific way, the being appears to have two cores. Three eyes, two cores -- same as Eva-01, even though the logo predates Eva-01's pseudo-evolution. In addition, Ritsuko states at Near Third Impact that the awakening Eva-01 is "returning to its original form" and "transforming into an energy condensate", which taken with the other points make Eva-01's true nature as an Adams all but stated outright.
Evangelion Mark.09 is sometimes considered a possible Adams, on account of being described as an "Adams' Vessel". However, the specificity of this label strongly suggests that Mark.09 is not an Adams per se, but, rather, is something intimately related to them, in a manner as yet indeterminate. Mysteriously, it is said to be "Wunder's true master", and is capable of wresting control of the ship away from Eva-01. Its entire body is a core exhibiting both metamorphic and imitative properties. In Evangelion lore, cores have classically served the function of a vessel: housing the resident soul and, when present, the S2 engine. This association could potentially shed light on what the Adams' Vessel truly is, but any such elucidation remains firmly in the realm of speculation at present.
The term "Adams" first appears in the preview for Eva 2.0 at the end of the first movie. In the next film, while the Adams make their first proper appearance during a stylized flashback to Second Impact, they go unnamed, as the sequence has no narration; nor are they named in the script, which refers to them only as only as "radiant giants" (光の巨人, hikari no kyojin). Even so, the name from the original preview was confirmed to still apply when the Evangelion 2.0 Complete Records Collection was published, wherein a concept sketch by Anno is captioned "Adams" (アダムス). The term "Adams" is finally used in spoken dialogue as of Eva 3.0.
Note that the term "Adams" is both singular and plural; "Adam" is not the singular form. While this sounds unnatural in English, Japanese does not require that the number of something be specified -- that is, no distinction between singular and plural is needed. It is usually assumed to be implicit in the context, or deemed otherwise unnecessary to mention. The pluralized form "Adams" was probably chosen for its ambiguity and provocativeness: the distinction from singular "Adam" immediately stands out, and prompts questions in the viewer about whether or not these "Adams" have altogether supplanted Adam's original role as the First Angel, or if, perhaps, there is more to it. A singular "Adam" (アダム) was mentioned in the Eva 2.0 screenplay draft published in the Complete Records Collection -- noteworthy because "Adams" were already established in the setting from the preview in 1.0 -- but this detail was altogether omitted in the final draft, replaced with nothing.
Eva-13 along with the two Evas strongly suspected of being modified Adams, Eva-01 and Mark.06, all possess forehead horns on their helmets. This motif also appears on Mark.09, which is an "Adams' Vessel" and not an Adams per se, along with the bridge of the Wunder, which is shaped much like an Eva helmet and evokes Mark.09's design specifically. Centered forehead horns are unknown for any other Evas, making this likely to be a deliberate visual motif. It is unclear if the horn motif has any greater significance beyond linking these entities together.
The shot of four impaled giants -- generally thought to be the Adams -- present in Until You Come to me at one point depicted some of the giants with horns as an actual anatomical feature (not merely a feature of the armor). However, the horns were removed in the process of genga revisions.
In Japanese mythology, a centered forehead horn is associated with oni, representing one of two common configurations that their horns can take. The oni connection has been acknowledged in the past, generalized to all Evas on a conceptual level but, in the final product, only truly evident with Eva-01.