There are many answers to this question within the field of psychology. Here are some of them.

From the World Health Organization (WHO):
"A state of well-being in which the individual realizes his or her own abilities, can cope with the normal stresses of life, can work productively and fruitfully, and is able to make a contribution to his or her community."

The absence of psychopathology (i.e., no psychological disorder). Lack of:
Deviance (i.e., culturally unacceptable behavior)
Distress (i.e., dysphoria)
Dysfunction (i.e., impairment)
Danger (i.e., violent behavior towards self or others)

The Positive Psychology movement defines mental health as the presence of psychological well-being. One such model is by Keyes & Lopez (2002) and includes:
High emotional well-being,
High psychological well-being,
High social well-being, and
Low mental illness
Another by Fredrickson & Losada (2005) describes "flourishing" as optimal human functioning composed of:
Goodness: happiness, contentment, and effective performance
Generativity: making life better for future generations
Growth: use of personal and social assets
Resilience: survival and growth after enduring a hardship

From Acceptance & Commitment Therapy:
Psychological flexibility

From the psychodynamic orientation, according to Shedler & Westen (2007):
Is able to use his/her talents, abilities, and energy effectively and productively.
Enjoys challenges; takes pleasure in accomplishing things.
Is capable of sustaining a meaningful love relationship characterized by genuine intimacy and caring.
Finds meaning in belonging and contributing to a larger community (e.g., organization, church, neighborhood).
Is able to find meaning and fulfillment in guiding, mentoring, or nurturing others.
Is empathic; is sensitive and responsive to other people’s needs and feelings.
Is able to assert him/herself effectively and appropriately when necessary.
Appreciates and responds to humor.
Is capable of hearing information that is emotionally threatening (i.e., that challenges cherished beliefs, perceptions, and self-perceptions) and can use and benefit from it.
Appears to have come to terms with painful experiences from the past; has found meaning in and grown from such experiences.
Is articulate; can express self well in words.
Has an active and satisfying sex life.
Appears comfortable and at ease in social situations.
Generally finds contentment and happiness in life’s activities.
Tends to express affect appropriate in quality and intensity to the situation at hand.
Has the capacity to recognize alternative viewpoints, even in matters that stir up strong feelings.
Has moral and ethical standards and strives to live up to them.
Is creative; is able to see things or approach problems in novel ways.
Tends to be conscientious and responsible.
Tends to be energetic and outgoing.
Is psychologically insightful; is able to understand self and others in subtle and sophisticated ways.
Is able to find meaning and satisfaction in the pursuit of long-term goals and ambitions.
Is able to form close and lasting friendships characterized by mutual support and sharing of experiences.