Non-religious meditation is often discussed as one of two basic procedures. One is single point meditation, where attention is kept on a single point. Whenever the meditator notices that attention has drifted into a story or a worry, that is dropped as a subject and attention is returned to the target. This discipline is continued for the available time.
The other method, sometimes more difficult for a beginner, is to simply watch the ideas and issues and themes and interests that enter the mind. The difficulty usually comes from dangerous or seductive themes or worries that latch onto the mind and engage it more deeply, worrying or planning, considering problems and possibilities and locking the mind into thinking.
Practicing, that is meditating regularly, as in a medical practice or a law practice, develops mindfulness, that is, awareness of what the mind comes up with and engaging that issue or direction deliberately or dismissing it. If a person couples meditation practice with reading and digesting the books "Incognito" by Eagelman and "Seven and a Half Lessons about the Brain" by Lisa Feldman Barrett, that person may develop greater awareness of the fact that the brain and the rest of the body are more, and do more all the time, than the conscious mind knows. Eagleman does a great job showing how our bodies digest food, circulate blood, breathe, balance and issue moods and impulses aside from our thinking minds. Barrett does a great job reminding her readers that "one's brain is not for thinking" - it's for heading up, being CEO, of the body.
You see this when you read the shock that scientists express when they carefully test the order of actions the body accomplishes. Say, I am going to reach for a peanut. My whole complex brain is involved after I consciously decide I want to reach for one, pick it up, and put it in my mouth. The brain issues a signal to my arm and hand to reach out, grasp a peanut, bring it to my mouth and insert it. Then, it sends a signal to my brain to take that action, and finally lets my mind know that I am going to do that.
Much research in many fields shows the advantages of practicing mediation, even for one minute a day.