How does Thich Nhat Hanh use Buddhist practice, especially mindfulness, to combat suffering?

Thích Nhất Hạnh is a Vietnamese spiritual leader and peace activist who is widely credited as a key figure in introducing Buddhism to the western world. 
How did he do this, and what are his primary teachings and beliefs?
He believes that mindfulness is essential for peace.
The only way for individuals to find peace and contentment, Nhất Hạnh believes, is to practice mindfulness (the practice of focusing one's attention on the present moment). This is often achieved through meditation. The only way for a society to function peacefully is for individuals to have inner peace. So mindfulness is very important in the prevention of conflict.
He encourages nonviolent conflict resolution.
Nhất Hạnh is a peace activist, and the peace he is trying to achieve is not only on an societal or political level, but also on an individual level (see above). He encourages individuals to proactively pursue peace and to set a good example for others. To quote the spiritual leader,

In Buddhism we see that teaching is done not only by talking, but also by living your own life. Your life is the teaching, is the message.

Nhất Hạnh's approach to nonviolence even includes animals: unlike human beings, animals cannot be taught to behave proactively or pursue peace, so Nhất Hạnh believes in protecting their interests. He promotes refraining from the consumption of meat and animal products as a way of practicing nonviolence to animals.
The points above are all leading up to the major point here:
Thích Nhất Hạnh practices "Engaged Buddhism." 
What is this? It is a term that Nhất Hạnh actually coined himself: it refers to the application of Buddhist principles, including meditation and mindfulness practice, to larger issues in the world—such as social and political issues, environmental problems, economic recessions, racial injustice, and so on. By working on ourselves first and by being aware of ourselves and the present moment we live in, we can be more effective members of society and even become leaders who set the tone for the communities we live in.