There is not simple distinction between these activities.  However:

Coaching tends to have a specific and tightly-focused goal (for example, helping someone prepare for an interview or make a presentation to a client group).

Training tends to be entirely work-related and concentrates exclusively on someon's professional skills and issues.

Mentoring goes further in offering support and advice to someone as a person and may touch on any aspect of their life.  The mentor may offer coaching or training from time to time but may also encourage the mentee to seek help from others in these roles.

*Taken from "The Mentoring Pocketbook", page 21
By Mary Crooks, Family Life Field Specialist