Mental health is the condition of having a mind that is "healthy" and functioning "well". I put these terms in scare quotes because there is no widely-accepted definition of what constitutes "mental health". In practical terms, the definition of "mental health" used by most of the medical community is the absence of any diagnosable mental disorder. There is considerable discussion among psychologists and psychiatrists as to the adequacy of this definition, and many alternative proposals have been made, such as mental health as a condition of overall well-being, or mental health as having rational beliefs and attitudes, or mental health as a proper balance of neurotransmitters in the brain; none of these have gained wide acceptance, and all of them seem to have important exceptions.
On the other hand, we also don't really have a clear definition of "physical health", and this hasn't stopped us from making a great deal of progress in medicine.
Mental hygiene is the process of working to maintain mental health. It involves such things as psychotherapy, medication, and even less clinical actions such as learning to meditate or reaching out for family support.
Mental health is the goal we are trying to achieve; mental hygiene is the means by which we seek that goal.
For example, one of the most common mental health problems is depression; appropriate mental hygiene to treat depression might involve cognitive-behavioral therapy or taking SSRI medication.