Sri Ramana taught that the individual self is an unreal, imaginary entity that persists because we never properly investigate its true nature. The sense of 'I', the feeling of being a particular person who inhabits a particular body, only persists because we continuously identify ourselves with thoughts, beliefs, emotions, objects, and so on. The 'I' never stands alone by itself; it always exists in association: 'I am John,' 'I am angry,' 'I am a lawyer,' 'I am a woman,' etc. These identifications are automatic and unconscious. We don't make them through volition on a moment-to-moment basis. They are just the unchallenged assumptions that lie behind all our experiences and habits. Sri Ramana asks us to disentangle ourselves from all these associations by putting full attention on the subject 'I', and in doing so, prevent it from attaching itself to any ideas, beliefs, thoughts and emotions that come its way.
The classic way of doing this is to start with some experienced feeling or thought. I may be thinking about what I am going to eat for dinner, for example. So, I ask myself, 'Who is anticipating dinner?' and the answer, whether you express it or not, is 'I am'. Then you ask yourself, 'Who am I? Who or what is this ''I'' that is waiting for its next meal?' This is not an invitation to undertake an intellectual analysis of what is going on in the mind; it is instead a device for transferring attention from the object of thought - the forthcoming dinner - to the subject, the person who is having that particular thought. In that moment simply abide as the 'I' itself and try to experience subjectively what it is when it is shorn of all identifications and associations with things and thoughts. It will be a fleeting moment for most people because it is the nature of the mind to keep itself busy. You will soon find yourself in a new train of thought, a new series of associations. Each time this happens, ask yourself, 'Who is daydreaming?' 'Who is worried about her doctor's bill?' 'Who is thinking about the weather?' and so on. The answer in each case will be 'I'. Hold onto that experience of the unassociated 'I' for as long as you can. Watch how it arises, and, more importantly, watch where it subsides to when there are no thoughts to engage with.
This is the next stage of the inquiry. If you can isolate the feeling of 'I' from all the things that it habitually attaches itself to, you will discover that it starts to disappear. As it subsides and becomes more and more attenuated, one begins to experience the emanations of peace and joy that are, in reality, your own natural state. You don't normally experience these because your busy mind keeps them covered up, but they are there all the time, and when you begin to switch the mind off, that's what you experience.
It's a kind of mental archaeology. The gold, the treasure, the inherent happiness of your own true state, is in there, waiting for you, but you don't look for it. You are not even aware of it, because all you see, all you know, are the layers that have accumulated on top of it. Your digging tool is this continuous awareness of 'I'. It takes you away from the thoughts, and back to your real Self, which is peace and happiness. Sri Ramana once compared this process to a dog that holds onto the scent of its master in order to track him down. Following the unattached 'I' will take you home, back to the place where no individual 'I' has ever existed.
This is self-inquiry, and this is the method by which it should be practiced. Hold on to the sense of 'I', and whenever you get distracted by other things revert to it again. I should mention that this was not something that Sri Ramana said should be done as a meditation practice. It is something that should be going on inside you all the time, irrespective of what the body is doing.
Though Sri Ramana said that this was the most effective tool for realizing the Self, it must be said that very few people actually achieved this goal. For most of us the mind is just too stubborn to be overcome by this or any other technique. However, the effort put into self-inquiry is never wasted. In fact, it's a win-win situation for most people; either you get enlightened, or you just get peaceful and happy.