To boast is to speak in praise of yourself. For example, a man is boasting when he talks about how great he is, the wonderful things he can do, or the things he has. He seems to think these things make him better than others. You can boast of something or about something, or boast that you can do something. The meaning doesn't change.
Ex: John was a very good athlete, but when he boasted about his sporting skills, it made his brothers very angry, because they knew he had more opportunity to practice than they did.
Ex: Mary was proud that she had a good paying job and earned enough money to buy a wonderful wardrobe, so she often boasted about the many beautiful dresses she had in her closet.
Someone who boasts can be called boastful. Boastful always refers to how someone speaks, not just how they feel. If I am proud of my singing ability, but never tell anyone about it, that's not being boastful.
An arrogant person is one whose pride is so great that it's annoying. This kind of person is said to be "full of himself". He thinks much of himself and not much about other people.
Ex: The boss tried to give Michael some helpful suggestions on how to sell more cars, but he is so arrogant that he will not listen to anyone's advice.
You can also speak of someone's arrogance, which is an insulting way of thinking or behaving which comes from people believing they are better or more important than other people.
Ex: Patricia has such arrogance about her medical knowledge! She thinks that, just because she took a year of nursing classes, she knows more than her doctor.