【Boast】 , 【brag】 , 【vaunt】 , 【crow】 , 【gasconade】 mean to give  vent in speech to one's pride in oneself or something (as family, connections, race, or accomplishments) intimately connected with oneself.

【Boast】 and 【vaunt】 are often used transitively as well as intransitively; the other words are chiefly intransitive.

【Boast】 is the general term; it may or may not carry a suggestion of contempt or impute exaggeration, ostentation, or vaingloriousness to the boaster.

【Brag】 is more forceful than 【boast】 and carries a stronger implication of exaggeration and conceit; it often also implies glorying in one's superiority or in what one can do as well as in what one is, or has, or has done.

【Vaunt】 is more literary than either of the preceding terms; it usually connotes more pomp and bombast than 【boast】 and less crudeness or naïveté than  【brag】 .

【Crow】 usually implies exultant boasting or especially blatant bragging in a manner suggestive of the triumphal crowing of a cock.

【Gasconade】 is the least common of these terms and implies habitual or extravagant self-vaunting.