Nightcore is a style of Electronic Music, in particular, a style of making music remixes where the song's pitch is altered (often to a major key) and the track is sped up to double the original BPM. Beats are made louder, and other edits are made, such as adding various digital effects, arpeggiated supersaw leads and eurobeat-esque backing melodies but the core element here is the speeding up. Basically a creative (ab)use of the Speedy Techno Remake, and user-made tracks are usually varied in quality, from well-composed, professional-sounding remixes to crappily sped-up songs with vocals that sound as if the singer had inhaled helium. There is also a unique type of Nightcore that is called switching vocals. There is 2 or more vocals mixed together with amazing effects. You can find many of these videos on youtube. Example Video: https://www.youtube.com/watch?v=dAiU_D0Ro8I
It's more of a phenomenon than a genre of its own, initially developing from the UK hardcore scene and branching out into other EDN genres from there. Most modern, well-made nightcore remixes fall under happy hardcore, UK hardcore, or eurobeat, though trance and even Drum and Bass-style remixes are not unheard of.
It has also become a catch-all term for any song that is sped up, which has become a source of Broken Base among fans of the style, as in such case, it's technically inaccurate simply speeding up a song without remixing it doesn't make it a "true" nightcore remix, with many Youtube content owners and practicing music producers calling out the uploaders on this.