"Normality"
I use what I think would be Carl Rogers's definition. For him, normality would probably mean self-actualization. This is a term coined by Abraham Maslow. It is one of the needs that emerge in human beings after their other more basic needs are met (see also http://www.wynja.com/personality/needs.html). It is sort of the "pinnacle" of human development. Maslow describes self-actualization as follows:
"A musician must make music, an artist must paint, a poet must write, if he is to be at peace with himself. What a man can be, he must be. This is the need we may call self-actualization ... It refers to man's desire for fulfillment, namely to the tendency for him to become actually in what he is potentially: to become everything that one is capable of becoming ..." (from http://www.performance-unlimited.com/samain.htm).
(To learn about some of the qualities of the self actualized person (actually, the process is always ongoing, so I should say the "self actualizing" person, see http://mind-brain.com/personality/sa.php and http://www.personcentered.com/selfact.html.)
Self-actualization itself implies an absence of anything that is significantly* interfering with the process of you becoming all that you can be. What can interfere with self-actualization (and, hence, normalcy?). Various psychological issues, which translate into non- or poorly adaptive defensive mechanisms, which in turn translate into serious mental disorders if the defenses become too severe or the issues giving rise to them are not resolved sometime soon (again, see 5 May 2003 and 6 May 2003).
*Note: We all experience some difficulty in self actualisation, not only because the process is ongoing, but also because our society does not always provide the conditions needed for this kind of growth. However, this mild difficulty in actualisation does not constitute abnormality. Only when there is a significant thwarting of the actualisation—and when that thwarting is due more to internal, personal psychological difficulties than to broader social circumstances (eg, poverty)—can we say that abnormality exists.
More generally, it could be said that "abnormality" is any thought, perception, conflict, behaviour, etc. that is not adaptive.
For more information on the distinction between "normality" and "abnormality", as well as such terms as "disorder", "neurosis", "psychosis", and "insanity", see the "Spectrum of Psychological Disturbance" in the second part of my Quick Reference Psychology Lists and Charts.
"Self"
One's essence. The inner, deep-down conviction of who one is as a unique individual—of who one feels one really is. This conviction includes—and is expressed through—the total gestalt of one’s cognitions (including attitudes), perceptions, consciousness, subjective interpretations, personality (including ego), self evaluation, and self esteem.
My definition of self is more of an individualistic one; it refers to the inner self—the part that is relatively untouched by external forces (if, indeed, such a thing exists...and I think it does). There are certainly social influences which determine a person's sense of self, but for my purposes, I'm not too concerned with those.
This differs from "identity" which refers to how you wish others to perceive you as an individual.
Does this work for anyone?