In theory, the objective of high-end audio is to reproduce the sound on a recording as accurately as possible. Hence, if the recording has a "warm" quality, that is what should be reproduced by a good system. Likewise, if a recording has a rather cool or analytical quality, that's what should be heard.
People being human, however, there are different "tastes" in reproduced sound. Some folks really like detailed, analytical sound (which, to my ear, often does not sound realistic), while others want to feel enveloped by warm, inviting sound. These preferences for sound quality used to be associated with the "solid state" camp versus the "tube" camp, but the best high-end audio gear today offers the best of both. I think that the term "warm" is used today to suggest the kind of sound that has some emphasis on the upper bass and mid-range, which is the portion of the audible sound spectrum where the human ear is the most sensitive (and, not so coincidentally, the range of the human voice).
People being human, however, there are different "tastes" in reproduced sound. Some folks really like detailed, analytical sound (which, to my ear, often does not sound realistic), while others want to feel enveloped by warm, inviting sound. These preferences for sound quality used to be associated with the "solid state" camp versus the "tube" camp, but the best high-end audio gear today offers the best of both. I think that the term "warm" is used today to suggest the kind of sound that has some emphasis on the upper bass and mid-range, which is the portion of the audible sound spectrum where the human ear is the most sensitive (and, not so coincidentally, the range of the human voice).