In its simplest terms, hi-res audio tends to refer to music files that have a higher sampling frequency and/or bit depth than CD, which is specified at 16-bit/44.1kHz.
Sampling frequency (or sample rate) refers to the number of times samples of the signal are taken per second during the analogue-to-digital conversion process. The more bits there are, the more accurately the signal can be measured in the first instance, so going 16bit to 24bit can deliver a noticeable leap in quality. Hi-res audio files usually use a sampling frequency of 96kHz or 192kHz at 24bit. You can also have 88.2kHz and 176.4kHz files too.
Hi-res audio does come with a downside though: file size. A hi-res file can typically be tens of megabytes in size, and a few tracks can quickly eat up the storage on your device or be cumbersome to stream over your wi-fi or mobile network. Thankfully, storage is much cheaper than it used to be, so it's easier to get higher-capacity devices. And technologies such as
MQA (see below) have arrived to help tackle that.