—Venerable Khenpo Rinpoche
The most basic form of Buddhist meditation is called "Calm Abiding" or "Shamatha" in Sanskrit, and is intended to rest the mind so that our primordial wisdom can shine through. Normally our minds are like "'monkey minds", swinging from one thought to another with very little rest in between. With Calm Abiding meditation practice, the mind can slow down and we can begin to become aware of our natural wakefulness. We can free ourselves from the tyranny of our thoughts and emotions so that our natural awareness can arise.
One can do this with single-pointed concentration by focusing on the breath. Sitting in a relaxed, straight posture, bring your attention to the breath. Focus on the sensation right at the nostrils as the breath comes in and out. If you are feeling agitated or a little excited, focus on the rise and fall of your belly. In any case, bring about 25% of your attention to the breath and let the rest just be spacious. When a thought arises, don't suppress it, don't follow it, and don't feed it. Just be aware that you are thinking and then bring your attention back to your breath without any further self commentary.
This is a practice of training the mind to keep coming back one-pointedly, so don't be discouraged if you keep getting distracted. This IS the practice in the beginning - to keep bringing the attention back to one point, time after time. If it helps you focus in the beginning, you can count your out-breaths from one to ten, then start over and count again.
Once you have stabilized you can stop the counting and focus only on the breath in a relaxed way, consciously realizing your inhalation and exhalation. Keep being aware as thoughts arise then return the mind to the breath. During your meditation sitting times, this may be as far as you get, and this is very good.
At some point as you continue to practice, your mind will begin to calm and awareness will start to arise. You may find that your thoughts have slowed down, or you may find that your thoughts are still coming, but you will be consistently aware of them and able to bring your focus back. This, too, is awareness. You may find that gaps open up between thoughts. When this happens, relax in the gap. With practice, you can keep making this pause even longer.
In any case, you will be less compelled by your thoughts and emotions. They may still arise, but you won't cling to them anymore. At this point, one can drop all focus of meditation, when you're ready, and let your mind merge with space. In other words, relax into the spaciousness of your true nature and allow the natural clarity to come forth. Rest in this natural state.