In Freudian psychoanalytic theory, the "ego" is part of one's personality that focuses on fulfilling desires while taking the real world into consideration.
Sometimes this means setting aside one's knee-jerk, intuitive drives in order to optimize for long-term gains (not buying a snazzy, expensive jacket in order to save for the down-payment on a house) and sometimes it involves concealing such behaviors with rationalizations that make beneficial investments and actions seem more immediately self-serving (refocusing one's attention on another, smaller purchase which then saves money for that longer-term goal of buying a house).
Egoism is a collection of philosophies oriented around how our sense of self—and in some cases the centrality of "self" over other considerations—shapes our behaviors, ethics, and actions.
Sometimes this means describing what we do in terms of how it benefits our wants or needs (the pursuit of self-interested outcomes) and sometimes it's more focused on what we should do to achieve those ends compared to what we actually do.
Such exploration can serve as a lens through which to view one's own actions in the context of self-gain and responsibility, but it can also serve as a method of self-censure. This has been especially true in the context of religious explorations of this concept: basically, how might I subtract my selfish internal needs from my behavior and refocus my attention on other, more moral (by my faith-inherited standards) pursuits (even if that means sacrifice and self-denial)?
Egotism is a drive to maintain a positive self-image, and to do all sorts of cognitive gymnastics (mostly justification, but also other perceptual distortions) to sustain a self-image that allows us to feel good about ourselves.
This is accomplished via all sorts of biases and misperceptions and overestimations, and many of our most difficult-to-counter (or even notice) perceptual issues can be tied to this need to feel powerful, valuable, worthwhile, and legitimate.
Egotism might be described as a lesser version of narcissism, though it could be argued some level of egotism is necessary for us to maintain a healthy, positive self-image and thus, a positive overall valence.
Egocentrism refers to an inability to understand the perspectives of others.
Said another way: someone who is an absolute egocentrist believes that everyone else sees things the same way they see things and is unable to comprehend that others might have other views, morals, experiences, and life paths from theirs.
Some people retain a more absolutist egocentric viewpoint throughout their lives (and often require some kind of psychological treatment to help manage it), but all of us experience aspects of this other-blindness—which is part of why we have so many prejudices and biases baked into our sense of reality to begin with—and it's more present at different points in our lives (infants and adolescents in particular have trouble conceiving of other people as anything but extensions of themselves).
As with many perceptual lens-related issues, adults tend to get better at countering egocentrism over the course of their lives not because they no longer see things through that lens, but because they've accumulated more evidence of its existence (and its downsides) and thus are more likely to counter it as part of their default perceptual routine.
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