1. Mono Check:
Mono playback is less common than it used to be. Most listeners will most likely hear your track in stereo if they are listening with headphones or on stereo systems in their cars or at home. Clubs, public systems and radio alarm clocks will however still play a summed mono version of your track.
Mono compatibility is however not the only or even the most important reason why you want to check your mix in mono.
The way your mix sounds in mono can be a good indicator of the quality of your stereo image. In other words, a solid stereo image will sound pretty decent in mono while a mix with a problematic stereo image will lose a lot when summed to mono.
People who hear your mix on stereo systems, outside of the sweet spot, where they usually are, will be listening to a slightly summed result due to cross-talk from each speaker reaching the other ear.
The main thing you want to check for is phase issues, which become very clear in mono.
It's always a good idea to do a quick mono mix check before your final bounce down, even if you set your balances in mono earlier on in the mix process. So, switch to your mono speaker or switch off one of your stereo monitors and use a plug-in on your master channel or the mono switch on your desk to sum to mono.
Listen to the entire track and make sure nothing disappears or jumps out at you. Pay attention to the relative levels between your instruments.
You don't want to obsess over mono. It's a useful tool to get a different context when you mix check and it helps you spot obvious issues with your stereo image. Your final mix will be a stereo track, so the best way to experience it will always be in the sweet spot on a good stereo system and in a good room. A quick mono check is more than enough.