If you check out the Archive, you will find a feature giving an in-depth answer to your question about record producers.
The mixing engineer oversees the technical operation of mixing the various recorded tracks into the finished stereo sound, usually acting on the producer's instructions to ensure that volumes are mixed correctly to give the chosen overall sound.
If a track has 'good production' it usually means that the producer has used his skill to obtain the best sound for the track - both in terms of technical production - correct mixing and so on, and also in terms of the arrangement of the track, maybe adding harmonies or orchestral backing if the track benefits from such additions. It is a term often used by people who don't necessarily know the technical operation of a recording studio, but they like what they hear as the final result.