In our everyday conversation how we can use these two words "boast" and "brag" and which one of them is taken in a negative sense?.Need some examples to differentiate between these two.
I really can't see a difference between the two, other than how they sound, and how they relate to various other words:
boast (n) | boast (v) | boastful (adj)
brag (n) | brag (v) | braggart (someone who excessively brags) | braggadocio (excessively boastful behavior, often considered to be mere artifice)
Even the dictionary uses one to define the other, so I would consider them more or less perfect synonyms.
Note: @aml suggests that "boast" is used more often for future accomplishments, and "brag" for past accomplishments, but I don't agree. I can't think of of a single context where one couldn't be swapped for the other:
He boasted/bragged of his wealth
He boasted/bragged that he would be rich someday.