There are two main forms of meditations, with a pretty much infinite number of variations: Concentration based meditation, and contemplative meditation. But it all bleeds together - contemplative practices require a certain level of concentration in order to be able to practice them.
Contemplative practices can range from meditation on a specific subject to general observation of your mind, thoughs, body, environment - this latter form is exemplified very well by mindfulness meditation (vipassana, from Theravada Buddhism, though when mindfulness meditation is taught in the West it is often pretty much cleansed of Buddhist material).
What the article describes sounds close to mindfulness meditation. In this case the goal is to quiet the mind, but not silence it or shut out external impression. The breath is used as an "anchor" that you return to in order to not let thoughts or feelings or observations drag you along, so that you can observe your thoughts in a detached manner, notice them, and move back to the breath.
Compare with some of the concentration practices which include focus on completely emptying your mind for extended periods of time.
Contemplative practices can range from meditation on a specific subject to general observation of your mind, thoughs, body, environment - this latter form is exemplified very well by mindfulness meditation (vipassana, from Theravada Buddhism, though when mindfulness meditation is taught in the West it is often pretty much cleansed of Buddhist material).
What the article describes sounds close to mindfulness meditation. In this case the goal is to quiet the mind, but not silence it or shut out external impression. The breath is used as an "anchor" that you return to in order to not let thoughts or feelings or observations drag you along, so that you can observe your thoughts in a detached manner, notice them, and move back to the breath.
Compare with some of the concentration practices which include focus on completely emptying your mind for extended periods of time.