The abilities they have differ from person to person and don't seem to fall into any designated categories. It is possible, however, to have the same power as another NEXT. NEXTs can't be identified unless their powers awaken.
They first appeared 45 years before the start of the series, due to an unknown mutation.
Awaken is a term used for when a NEXT's powers first start to emerge. Most NEXT can be identified by the blue glow that surrounds their body when their powers are active - as well as their eyes taking on the same blue hue. However, this blue color is apparently not the only indication of a NEXT as both Jake and Kriem glowed reddish-orange when using their NEXT abilities. Additionally, once Kotetsu's powers started dwindling, he would occasionally take on a reddish-orange glow as well.
Some exceptions exist, however. Origami Cyclone only gives off a blue glow when he is in the process of transforming into someone. While transformed, he does not glow at all. Lionel Hawke doesn't glow when his NEXT powers are active, either. Instead, particles of blue light disperse around him as he activates/deactivates his power. Lunatic also doesn't seem glow when his powers are active, though often his eyes give off the same sort of flames his power creates. Additionally, Jake doesn't glow when using his Hearing Others' Thoughts ability, as it is always active. Most hero suits are equipped with a "Lighting Mode" function, whereby part of the suit will light up when the user's powers are active.
There are a wide variety of NEXT powers, though not all serve a practical purpose (Super Sweat, Stretchable Skin, etc). Some come with a certain condition or time limit - such as Hundred Power's 5-minute limit and hour-long cooling off period or Lionel's invisibility lasting only as long as he holds his breath.
For a complete list of powers in the series, see the NEXT Powers.
Hero Academy is a school operated by Timo Massini where all NEXTs are trained to become heroes. Barnaby, Ivan and Edward all attended, though Kotetsu did not - which could imply that school didn't exist when he was growing up over the years. The school seems to teach and train NEXTs to control their powers effectively and ultimately become future heroes. Students are expected to use their powers only on school grounds while attending, and it would seem that NEXT of all ages are welcome. Most of the Hero TV heroes did not attend Hero Academy, so how enrollment functions is currently unknown.
Currently, only one NEXT has been shown to have more than one power at the same time. The phenomenon is extremely rare, and might even have been considered impossible prior to Jake's revealing his dual abilities as none of the heroes had heard of such a thing happening before.
In rare instances, a NEXT's powers can gradually dwindle down until vanishing entirely. Information regarding the phenomenon is so scarce that Ben had to go digging around quite a bit before finding something significant. How or why this happens is still unknown and only two cases exist at present: those of Wild Tiger and Mr. Legend. It may be due to having a child, as both Tiger and Legend do have a child: Kaede and Lunatic.
In Wild Tiger's case, his Hundred Power deactivates earlier than the designated 5-minute span. Each successive activation of his power shaves off a few seconds of activation time. Though his power can't be used for the usual duration, its effects are dramatically increased for the span it is active. Additionally, sometimes Wild Tiger's usual blue glow changes to a reddish-orange right before his power times out. It remains unknown if this is an indication of his powers fading or of him developing a second one.