It’s somewhat of an elevated midbass region that adds a lot of body to most instruments. Especially on something like keyboards, synths with a bit of low end, synth bass, a warm system will make those sound extra pleasant and rounded. If it’s too much the whole mix gets bloated. Also very obvious on guitar work and other stringed instruments, if you lack ‘warmth’ you will have those sounding sterile and unsatisfying. Some genres like certain metal subgenres do prefer this sound, with shriller guitars and drums and a lot of high end instead. It sounds very aggressive but somewhat thin.

This is pretty much the context I’d use it when mixing or mastering but I think it applies exactly the same when casually listening.

Reading some of the comments I personally disagree that a warm sound means rolled off highs; it’s perfectly possible to get a warm mix and not cut out the high end as long as you leave enough room for the instruments to breathe. I personally don’t like headphones or speakers that are actually quite dark - the high end needs to be there and when it’s done right it’s the most intoxicating thing to listen to as long as the instruments have enough texture from that ‘warmth’ at the same time