boast
To brag about yourself or something you did.
To boast means to talk about yourself or your achievements in a way that shows off, often making others feel like you think you're better than they are. When someone boasts about winning first place in the science fair, they keep talking about it to make sure everyone knows how smart or special they are, dwelling on their success to impress others.
There's a difference between stating a fact and boasting. Saying “I got an A on the test” is just information, but saying “I aced that test because I'm basically a genius” is boasting. The boaster wants to impress others and make themselves look superior.
Sometimes people boast because they're insecure and want others to think highly of them. Other times they genuinely don't realize how their words sound. Either way, boasting tends to push people away rather than impress them. Your actual friends care more about who you are than what you can brag about.
You can also use boast in a more neutral way to describe something a place or thing has: “The library boasts more than 50,000 books” just means it has that many books and can be proud of them. But when talking about people, boasting usually carries that sense of showing off.