These three verbs describe situations in which someone or something is moved from one place to another. The way they are used depends on where the speaker is with regard to the listener and vice versa.
Bring and take
Julie tells a friend that she invited John for dinner:
John came round for dinner last Saturday. He brought me some wine and a beautiful bouquet of flowers. (From Julie’s viewpoint, she is the destination of the movement.)
But John would say:
I went round to Julie’s on Saturday. I took her some wine and some flowers. (From John’s viewpoint, he is responsible for the movement.)
Fetch means to go to a place to get, buy, or collect something or someone and return:
Take a seat and I’ll fetch you a glass of water.