An egoist is someone who thinks he is much better or more important than other people.
Mr. Cumberbatch is such an egoist that no one can work with him. Many of his employees have quit because he always has to have things done his way.
Although the musician is considered a master and has performed around the world, she is not an egoist. She teaches young people how to play and encourages them in their dreams.
Egoist comes from the Latin word ego meaning "I" or "self". The phrase alter ego means "another self". This phrase is used for someone who acts as a substitute or it can be used to describe a close friend.
Mr. Jefferson said he could not make the meeting, but he would send his alter ego, Mr. Washington, who had full authority to act in his place.
Philomena, it seems, cannot go anywhere without her best friend and alter ego, Emmeline; they have been friends since kindergarten and it seems they always will be.
Conceited means having too much pride in your own worth or goodness or being full of not necessary pride. A conceited person seems to think that he or she is better than others, when they have little reason to think so.
Alexa is difficult to talk to because she's so conceited about her intelligence. She's not really as clever as she thinks she is; I know many people who are more intelligent than her.
Conceit is not necessary pride in yourself or your accomplishments. Another use of conceit is for a fanciful or creative idea.
Nathaniel's conceit for the film was an amazing fantasy world of the future where men fight battles with mythical creatures and win the love of beautiful ladies.