Negotiable Instrument Act, 1881

You Get a stipend check of Rs. 1,000/- from ur Boss.

now the check is an instruction to bank to pay you onbehalf of him.

in this case

ur sir,-who make check & order to pay you is become Drawer

You- You will get the payment & you r going to be paid so u become payee

& bank to whom order is made become drawee.

ok.

 Please tell me in context of bills of Exchange. Who Draws the bill and the signing authority. 

In the case of Bill of Exhange the person who draws the bill is the drawer and the person who accepts the bill is the Drawee and the payment to the banker is the payee