There is a widely spread misunderstanding of these three terms. Often they are used interchangeably thus a negative shadow has been cast over all of them. Egocentrism is a childish (nevertheless it could frequently be observed in adults behavior) state of mind in which one is unable to accurately assume or understand an opinion or perspective different from one's own; egotism is defined as the drive or tendency to overestimate one's own personality, qualities, and values; whereas egoism stands for the philosophical presumption of the profound importance of individualism usually opposed to the moral censure of self-interest supported by the more conservative and dogmatic parts of society. Therefore, the first and the second are somehow deviations from the clear perception of one's own place in the world, while the third is the acceptance of the personal rights that each individual embody. In that regard, egoism is to principally base your convictions on the belief that the most genuine and true source of knowledge for one is the individual's experience of world phenomena.
Of course, preoccupation exceptionally with one's own personality and beliefs could end up in egocentric or egotistic behavior. However, egoism is a precondition for, on the one hand, the recognition of the Other as an individual and thus as someone worthy of consideration, and on the other hand, for the inner drive for personal development and improvement. After all, the best that one could give to the world is his unique point of view and talents. Or, as Oscar Wilde might say – playing someone else role is boring, better play yourself!
"Be yourself; everyone else is already taken."
― Oscar Wilde