Breath meditation is called ānāpānasati in Pali. It is one of the 40 techniques for "tranquility meditation" in Buddhism.
Main Instructions: These are given on the first day of the 10-day course: sit with your back straight, close your eyes and "pay attention to your natural breath in the nose region". The instructions are deceptively simple.
Ancillary Instructions and Guidelines: Along with the main instructions, several ancillary instructions and guidelines are given for breath meditation, especially during the evening discourses. For example, one guideline is that if you feel sleepy, you may get up and even take a short walk to ward off sleep. Another guideline is to take off spectacles to avoid the tendency to open one's eyes. Another guideline is to avoid overeating during breakfast and lunch because it is difficult to meditate with a full stomach; one should keep the stomach one-fourth empty. Another guideline is that sitting straight would be important in the long run; however it is not a major impediment to progress; chairs, cushions and back rests are provided in the meditation hall for anybody to use. And so on.
It is important to follow such instructions and guidelines because these have been developed for good reasons. Many of these guidelines have emerged in response to frequently asked questions by students. Still, each student has questions that are unique to his or her own situation, for example, due to health reasons and due to lack of clarity about what some word or some instruction really means. Assistant teachers are available to answer such questions.
Posture (sit with your back straight): Getting one's back perfectly straight is a tall order. S N Goenka says that a straight back would 'help in the long run'. However, it is not an impediment to progress, at least for beginning students.
Acquiring a comfortable posture is important because one has to sit for long hours. However, discovery of a comfortable posture might require some experimentation and time. At the 10-day camp, no specific posture is recommended or encouraged; you may sit in any posture that is comfortable for you. Some people cannot sit on the floor and are comfortable in only a chair. Some sit on the floor in vajrasana, others sit in padmasana. Many who sit cross-legged on the floor find it helpful to place a cushion to raise their tailbones. Chairs, pillows, cushions and meditation stools are available to help you figure out the combination that works best for you.
You shall soon realize that you cannot sit for an hour with your back straight with tight muscles everywhere. And that is the beauty of meditation: the only sitting posture that allows you to sit motionless for extended periods of time is one where your muscles are relaxed and your spine is straight, both at the same time. Regular daily practice of exercises like Yoga/Pranayama from India or Qi Gong/Tai Chi from China helps improve posture.
Breath in the nose region: In Chinese practices like Qi Gong, they say that the entire body breathes. For example, the whole body of an infant gets inflated and deflated with every breath quite visibly. So breath can be observed at many parts of our body, especially in the abdomen which makes large movements. In fact, in the Mahasi Sayadaw tradition, one starts off by observing the movements of the abdomen (see instructions for meditation). However, at the 10-day camp, the instructions are to observe the breath in just the nose region, and to pay no attention to other parts of the body.
Natural breath: The instructions are to observe the breath as it naturally flows in and out. S N Goenka emphasizes repeatedly that the goal of meditation is to look at things as they really are (yathā-bhūta in Pali), and not as you would like them to be. So one should observe breath in its natural state. In contrast, breathing exercises like Pranayama modify breath. During meditation, we should not modify our breath. However, an ancillary instruction given at the course is to 'take a few hard breaths' if you are not being able to maintain attention to breath for long, and then to go back to natural breathing.
Paying attention: Paying attention to breath is the hardest part of the main instructions. Initially, you might discover that within two to three seconds of paying attention, your mind wanders away and you start thinking of something (some problem at work, some conversation with a colleague, some incident in your life, some person whom you like, and so on). After several seconds or minutes, you would realize that you actually sat down to observe your breath! And then, within two or three seconds of observing your breath, you start thinking again, forgetting that you have to pay continued attention to your breath. And the cycle continues. For this reason, it is important to have 1-hour sittings as a beginner because if you sit for only 10 minutes, you might have paid attention to your breath only two or three times for only two seconds each.
And when you were thinking, what were you thinking about? It turns out that thoughts occur in a sequence, successive thoughts are connected through some aspect. Using the lingo of computer scientists and mathematicians, there is a network (or graph) of thoughts, with two thoughts connected if they share something in common. And the mind hops from thought to thought along such a network. This is best explained using an example. You may remember your favorite movie, then you may remember that you had a conversation with a friend outside the theater, then you may remember that that friend has asthma, then you may remember that your grandma had asthma, then you may remember that your grandma gave you a sweater as present, then you may remember that you have a nice picture of yourself in that sweater, and so on. Usually, the mind works quite fast, and hops from thought to thought quite rapidly. The phrase "mind chatter" is often used to describe this phenomenon.
When you start observing your breath, when you return to breath, you might have memories of the last three or four thoughts -- analyzing these thoughts would help you verify that the 'network theory' of thoughts is actually correct! Such analysis of the last few thoughts is called 'Retracing' in the article (Note: most of this article is likely to make sense only to dedicated meditation practitioners) Experiments in Insight Meditation by Rod Bucknell, Australian Journal of Transpersonal Psychology, 3 (1983), 96-117. The author spent over four years practicing Vipassanā meditation in two different monasteries in Thailand, in the tradition of Mahasi Sayadaw. The 'Retracing' described in the article is not part of the instructions given at the 10-day course, and is not encouraged — it is a distraction.
Physical changes: Attendees like me have busy lives in cities, some of us do not exercise. My personal experience at the 10-day camp: day by day, the muscles in the abdomen, lower back, upper back, shoulders and arms became more and more relaxed. The litheness and relaxation was quite noticeable. My full-time job is sedentary; I sit in front of a computers for long hours every day. The neck and shoulder muscles of such professionals are usually tight, especially if they do not exercise. The flexibility I felt in my muscles reminded me of machines that operate smoothly after applying lubricating oil. Such flexibility also improves posture and one can sit for longer hours in meditation.
Impact of sīla (morality) on samādhi (one-pointedness): Sīla and samādhi are said to be synergistic — they help each other. So progress in tranquility meditation is impeded by lack of moral conduct. I do not understand the mechanism clearly. My intuition is that if any of the precepts are broken, the mind gets preoccupied with thoughts related to the event where the precept was broken, and it is difficult to pay attention to breath.
Development of breath meditation: By the second day at the 10-day camp, students are asked to limit observing the flow of breath to the front portion of the nostrils, with emphasis on the 'touch of breath' as it goes in and goes out. You will start noticing that the in-breath is cold and that the out-breath is warm, and that typically, the out-breath is longer. The change from merely observing the flow of breath in the nose region to observing the touch of breath in the front portion of nostrils is significant. On the third day, the instructions are to observe sensations in the triangular region from the tip of the nose to the two ends of lips. The ability to observe sensations makes us ready to start vipassanā meditation. The word sensation requires explanations and is discussed in Part III of this article. By the third or fourth day, additional guidelines are given: we should sit as still as possible and make very slow movements to change posture when necessary.
How to measure progress? Progress in breath meditation is measured by two parameters: (a) the duration for which undivided attention can be paid to the touch of natural breath, and (b) the smallness of the region between the upper lip and the nostrils at which the touch of natural breath is felt. As progress is made, natural breath shall gradually become longer and lighter, its touch being barely perceptible. In one of the discourses on the fourth or fifth days, S N Goenka mentions that being able to pay attention for more than a minute denotes significant progress.
One-pointedness, concentration, relaxation and sharpness: The goal of breath meditation is to maintain undivided attention to the touch of natural breath for extended periods of time in as small an area as possible. It is in this sense that the mind becomes one-pointed. As the mind gets more and more one-pointed, breath becomes longer and subtler. So the touch of breath becomes progressively less perceptible. Being able to pay attention to subtler and subtler breath in the nose region requires a sharper and sharper mind -- it is in this sense that the mind is 'sharp'. Such sharpness shall be concomitant with more and more relaxation of the body. Soft tissues in various parts of the body shall loosen up as progress is made. Conversely, you will notice that when the mind is busy hopping from thought to thought, breath would be shorter and harder and the body shall not be as relaxed and the mind would not be as sharp. In a nutshell, long subtle breaths, one-pointedness, concentration, relaxation and sharpness are all correlated.
Further Reading: This article published by the Vedanta Society of California has a section on 'five states of mind'. One-pointedness (ekaggatā in Pali), the state of mind achieved by tranquility meditation, is called 'ekagram' (a Sanskrit word) in the article. Some who have never meditated tend to believe that one gets 'ekagram' when one is absorbed in painting or when one is listening to music or focused on their work. In psychology, the term flow, coined by positive psychologist Mihály Csíkszentmihályi is used to describe such experiences. My personal belief is that flow experiences are not exactly the state of 'ekagram' that one acquires through tranquility meditation, even though both are wonderful exercises for stress reduction. The state of 'ekagram' would most probably be a new experience for you, after progress is made in tranquility meditation.