We are all used to hearing songs after they are fully mastered. If a song wasn't mastered in a way that you like, you may feel it has too much treble, or not enough bass, or visa versa. Maybe you will feel like the song needs to be turned up louder than other music you listen to... or maybe you can't get the music loud enough in headphones. These issues would all be conditions of the mastering.
Once a mix is complete, the song is not ready for distribution because the mix only focuses on the overall balance of each instrument, and their role in said song. The mixing will create the environment, where mastering is the final polish to that environment. A good master will accentuate a good mix or even a bad mix! On the contrary, a bad master will ruin even the best sounding mix.
Mastering focuses on the overall harmonic balance of the songs frequencies and its "loudness". Some genres are louder than others, which is part of the feel of that specific genre. Something that's more emotional and soft might be ran though tape in order to create a "nostalgic" sound, while using other equipment to boost the signal musically in order to get to the standard volumes required by the playback devices and distributors. Other genres, might be the opposite, where they might sound loud and in-your-face intentionally. How mastering is accomplished widely varies, and it depends on the need of the specific song.
However, all songs are victim the LUFS (Loudness Units relative to Full Scale) standards set by the distributors of the music industry. Different platforms enjoy different volumes, so mastering a song to a standard is important so your song has a similar volume as the others on whatever listening platform is being used. If those LUFS standards are ignored while mastering, it's not likely your song will shine like the other professionally mastered songs on your playlist.
Because mastering is the final step to completing a song before its release, the quality of the mastering job is hugely important to the overall sound and potentially the success of any project.