The term "centering prayer" is most often associated with Christianity as it was popularized by Fr. Thomas Keating, the renowned Trappist monk and priest. To learn Thomas Keating's particular method for centering prayer, go here. You may also visit the main website for centering prayer.
People of all faiths or no faith can practice centering prayer, adapting the practice to their beliefs by choosing the prayer word. A good general prayer word is "Peace." There is no need to focus on a deity and an atheist may choose to focus on a deep feeling of shared humanity.
When I have practiced centering prayer, I have given myself anywhere from fifteen minutes to an hour, depending on the time I have. I like to light a candle to ritualize my sacred time. I recite the Lord's Prayer pausing in brief silence after each line,
and then I sit on a meditation cushion or a chair, with my back straight or supported. As per the traditional method, I use a prayer word, probably one of the most common ones for Christians, Maranatha which means "Come, Lord" in Aramaic. I breathe fully and peacefully, not straining the breath yet remaining aware of the in-breath and the out-breath. I silently repeat the prayer word to myself with each breath. After a few minutes of doing this, I allow myself to simply sit and welcome God's presence, however it may manifest itself to me. Often there is a feeling of peace and balanced openness to whatever comes. Any thoughts I have I embrace with compassion and they easily dissipate.
When you engage in this practice, know that you are joined by people all over the world and acknowledge your connection to them. In this way we all support one another.