To show off means to display yourself or your things in a way to attract another's attention. You could show off by telling how much money you make or show off your new car by driving it very slowly up and down the street to make sure everyone sees it. It can be used in a positive or a negative way.
Ex: Cecily got to show off her amazing salsa dance skills last night at the party, with her partner, Eric. We were all very impressed, and wondered where she took lessons.
Ex: Albert was really showing off when he told us how well he did in his classes. He was blessed with great natural intelligence, so it's not surprising he got all A's, but he shouldn't boast about it.
Show off can also be used a noun. A show off is a person who shows off a lot. This is always used in a negative way.
Ex: Greg is such a show off; he tries to impress every girl he meets, by telling them how much money he earns and what a big house he has.
A person who takes a lot of pride in how he looks or his abilities can be called vain. A vain person is someone who always wants others noticing him and admiring him. Margaret is enormously vain about her looks.
Ex: She carries a mirror with her and admires her beauty whenever she gets a chance; she ought to be humbler about it.
If a person is vain, he has a lot of vanity. Vanity means pride in yourself. Both vain and vanity are used in a negative way.
Ex: Harrison's vanity was crushed when he found out that Kevin earns more money for doing the same job as him, especially as Harrison has been at the company for longer.