Hi-resolution audio (also known as hi-res, high definition or HD audio) doesn't have an official definition - but it's generally accepted to mean any audio file stored at higher quality than CD.
Some experts argue that CD quality is plenty high enough for human hearing - but hi-res files certainly represent a big step up from standard MP3s or Spotify streaming at 320kbps. To put it into perspective, a CD has a data rate of 1411kbps (over four times that of an MP3 at 320kbps) while a hi-res music file has a data rate of 9216kbps - over 28 times faster (or more detailed) than the highest quality MP3.
This does however mean that hi-res files (including WAV, AIFF, FLAC, ALAC and more) will take up much more space in your phone's memory. So if you're thinking of going down the hi-res audio route, you'll want to choose a phone with plenty of memory. Plus of course, you'll want to pair it with a decent set of headphones.