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 the 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 glowing the same blue. However, this blue color is apparently not the only indication of a NEXT as both Jake Martinez 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 Telepathy ability, which is always 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 cooldown period or Lionel's invisibility lasting only as long as he holds his breath.
For a complete list of powers in the series, see NEXT Powers.
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 change. Information regarding the phenomenon is so scarce that Ben Jackson had to go digging around quite a bit before finding something significant. How or why this happens is unknown and only two cases are known to exist at present: those of Wild Tiger and Mr. Legend.
Mr. Legend lost his powers completely.
In Wild Tiger's case, his Hundred Power began to deactivate earlier than the designated 5-minute span. Each successive activation of his power shaved off a few seconds of activation time, until he was left with only one minute.
There may have been other cases however, as Ben tells Wild Tiger that he might get stronger, get weaker, lose his powers entirely, or even gain a new power.