The idea of the "self" is an incredibly important, yet complex concept. I'm sure that when I talk about "self" everyone has a vague notion of what I mean, but it is not exactly easy to put into words. Let us take a look at some of the more common psychological ideas surrounding the notion of self.
Self-concept: your idea (conception) of your persona. This includes the parts of your personality, skills, abilities, knowledge, etc. that you show the world and the parts that you keep hidden from others. When someone asks you to describe yourself, you are tapping into your self-concept.
Self-esteem: your overall evaluation of your self. You may have an overall positive view, overall negative view, or even a neutral view of your self. People often tie their self-esteem to many irrational things, such as the viewpoints of others. People even tie their value to unrealistic characteristics that are beyond their control--not everyone can be #1, but that somehow doesn't stop many people from trying to reach it.
Self-image/public self: the "face" you show to other people. Each and every one of us, even the nonconformists, cultivate an image that we project to others. Everyone wants to be accepted for who they are, though many individuals often try to achieve this acceptance by manipulating the face that others see. We hope to garner their respect by showing them what we think they want to see, rather than taking the risk of showing our real self.
Private self: the part of our persona that we keep hidden from others. We may reveal pieces of our private self to a select few. We typically hold back at least a small portion of our self, however, in order to hang onto at least a small amount of control over ourselves. Introverts tend to have a much larger portion of their self in this private segment than extroverts.
Real/true self: every facet of our persona; the good, the bad, the beautiful and the ugly. Our real self includes not only our strengths but also our weaknesses. This houses our "shadow," that darkness within, the flaws that make us human and the qualities that help us rise about our frailties. When we are completely honest with ourselves and others we can acknowledge our true self. We tend to spend a lot of time denying parts of our self, however, which could lead to doubts and/or depression.
Ideal self: the "ought" or "should" or desirable qualities that we think we need to possess. The ideal self comes from within and from others. Many parents tell their children that they should do this or that, they ought to behave in such a way or achieve certain goals in life. We run into psychological problems when our real self and our ideal self do not overlap significantly. Think of Dr. Zoidberg from Futurama. He grew up wanting to be a comedian, but his mother pushed him into medicine. One of the results was a lingering doubt, low self-esteem, and a perpetual unhappiness with his life. This is just a fictional character, but there are many real people out there who have been handed (or chosen for themselves) unattainable goals and they live with the reality that those goals will never be met.
Self-worth: the value that you place upon yourself in the context of a larger social setting, such as a distinct group of people or within society at large. Many people who seek their identity do so in order to increase their perceived self-worth within the reference groups of their choice.
Self-efficacy: your belief in your ability to accomplish a particular task. Self-efficacy is roughly equivalent to self-confidence and it is task-specific. We develop our notion of self-efficacy as we attempt new tasks, confronting the unknown. This does not necessarily affect our self-esteem. You may have low self-efficacy in spelling, for example, believing that you could not spell to save your life, but feel overall positively about yourself. The effect of self-efficacy on self-esteem is moderated by the importance we (and others whom we consider important to us) place upon tasks. For example, let's take a look at two different adolescents. Both of these teens are not very athletically inclined. For teen A, sports are not important as they have decided to pursue a life of academia. For teen B, sports mean the world because their parents place too much value on physical achievements. Teen A may feel embarrassed by their inabilities in gym when they are in gym, but overall they don't let it affect them negatively because it's seen as unimportant. For teen B, however, this lack of athletic skill is seen as a personal failure in life, an inability to meet the unrealistic demands of an unreachable ideal, so their self-esteem is bottomed out by their perceived failures.
Self disclosure: the process of sharing personal information about our self with others. Self disclosure begins simply at first--an exchange of names, telling someone what you do for a living--and gradually increases as the intimacy in the relationship grows. [Intimacy, in this case, is psychological closeness; it is a measure of how well you know a person and how well they know you.] Typical relationships start off with equal self-disclosure. One person shares a bit of information and the other reciprocates with the same level of disclosure. After a time is becomes natural for one person to share slightly more than the other. Sometimes a person will share too much deeply personal information too quickly and the relationship may become stunted. Sometimes one person will refuse to share any information about themselves and the relationship will stall.
Are there more "self" notions out there? Definitely. I will leave you to explore some of these ideas on your own. The pursuit to understand one's self, according to humanism, is a life-long journey. The more you learn about your self, the more you will discover that you do not know. In the end, it is up to each individual to decide how to define their personal self.