The Sanskrit word "mantra" literally means "instrument of thought," or "sound tool."  A mantra is a vocal sound, or set of words, that can be repeated (either to one's self, out loud, or both) in order to bring about a specific mental state. Mantras are used widely in yoga, Hindu, and Buddhist practices, and believed to have psychological or spiritual power. 
Mantras can be used during meditation practice to establish a desired mental state, and then link this mental state with the sound or set of words that are repeated during the meditation. When one is later in an undesired mental state, the mantra can help one return to the mental state linked with the mantra. For example, if I say the word "peace" repeatedly, and imagine sitting by the ocean, then later, if I am very anxious, saying the word "peace" several times can help me to think of the ocean, and then to relax.
Benefits of Mantras
Meditation using mantras can yield all of the benefits of mindfulness meditation, with added benefits. By training a specific mantra to yield a beneficial mental state, one can use it during a negative mental state to recall the beneficial state, and thereby to decrease negativity. How well this works is related to how much practice one has invested in using the mantra. A medical study done in Florence and Pavia Italy found that meditation using rosaries, or mantras positively and significantly influenced breathing rate, blood pressure and cerebral circulation.  Some traditional yoga practitioners believe that certain sounds produced by the vocal chords can resonate throughout the body, and that these resonances have a beneficial effect on the body. 
Choosing a Mantra
Good mantras tend to be short and easy to remember, consisting of a single sound (e.g. "om"), or between one and ten words. If the chosen mantra consists of a set of words (and not a sound), then the meaning of the words should be consistent with the mental state that is desired. For example, if I want my mantra to increase a feeling of peace, then a mantra like "I am full of peace" could work well.
How to Use Mantras
The following steps provide a guide about how to practice a mantra, in this case "I wander in fields of nothingness," and how to invest a specific state of mind in the mantra so that it may later be used to recall a peaceful state. Follow these steps (feel free to modify them however you like, and to choose a different set of words that are in keeping with your goals):
(1) Close your eyes if you feel comfortable doing so.
(2) Slowly breathe in through your nose and out through your mouth several times until you feel more relaxed.
(3) Focus on your breath.
(4) Keep breathing as above, and try to empty your mind.
(5) If thoughts pop into your head, just notice them, and then let them go.
(6) When you feel more focused, say "I wander in fields of nothingness". Say this to yourself, or out loud, whatever feels most comfortable.
(7) Imagine a large grassy field.
(8) Imagine that you are standing in that field.
(9) Touch the grass with your hand, and imagine how that feels.
(10) Take a step further into the field.
(11) Now say "I wander in fields of nothingness" again.
(12) Repeat from (6). Keep slowly repeating this, roughly 6 times per minute.
(13) When you feel at peace, and are ready to stop, focus on your breath again.
(14) When you are ready, open your eyes.
1. Whitney, William, "Sanskrit Grammar," Harvard University Press, Cambridge, MA, 1950.
2. Feuerstein, Georg, "The Deeper Dimension of Yoga," Shambala Publications, Boston, MA, 2003.
3. Bernardi, Luciano, et al, “Effect of rosary prayer and yoga mantras on autonomic cardiovascular rhythms: comparative study,” http://www.ncbi.nlm.nih.gov/pmc/articles/PMC61046/.
4. Axel, Gabriel, "Your Brain on Om: The Science of Mantra," http://health.usnews.com/health-news/blogs/eat-run/2013/10/02/your-brain-on-om-the-science-of-mantra.