The meaning of the word eutony is well-balanced tension.
Gerda Alexander, developer of this somatic practice, said that it is a Western way of experiencing unity of the whole person. She also said this unity frees your creative forces and develops your ability to make contact with others while maintaining your individuality.
The goal of Eutony is to widen and deepen the whole range of expression, open up creativity, and help you feel at home in your body. The work is gentle and never forces. Presence is of the upmost importance:
When doing this somatic practice you pay attention to how movements affect your perception of your body and how your tension and mood changes. Each movement influences the tone (tension) of both your muscles and your inner being. You develop movement awareness and also develop depth in yourself, growing toward being a total person.
Here's an interview with Gerda Alexander from the January/February 1986 issue of Yoga Journal.
"Interview with Gerda Alexander" by David Bersin in Bone, Breath, and Gesture: Practices of Embodiment, edited by Don Hanlon Johnson.
Discovering the Body's Wisdom by Mirka Knaster has a chapter on this somatic practice.
Eutony: The Holistic Discovery of the Total Person by Gerda Alexander.
Planet Medicine: Modalities by Richard Grossinger has a chapter on this practice.
Article documenting some classes: Eutony for all bodies
Videos on YouTube (In Spanish, but some of the visual information is enlightening, even if you don't understand Spanish.)