The Collaborative International Dictionary of English v.0.48:

drawee \draw*ee"\, n. (Law)
   The person on whom an order or bill of exchange is drawn; --
   the correlative of drawer.
   [1913 Webster]

WordNet (r) 3.0 (2006):

drawee
    n 1: the person (or bank) who is expected to pay a check or
         draft when it is presented for payment

Bouvier's Law Dictionary, Revised 6th Ed (1856):

DRAWEE. A person to whom a bill of exchange is addressed, and who is
requested to pay the amount of money therein mentioned.
     2. The drawee may be only one person, or there may be several persons.
The drawee may be a third person, or a man may draw a bill on himself. 18
Ves. jr. 69; Carth. 509; 1 Show. 163; 3 Burr. 1077.
     3. The drawee should accept or refuse to accept the bill at furthest
within twenty-four hours after presentment. 2 Smith's R. 243; 1 Ld. Raym.
281 Com. Dig. Merchant, F 6; Marius, 15; but it is said the holder is
entitled. to a definite answer if the mail go out in the meantime. Marius'
62. In case the bill has been left with the drawee for his acceptance, he
will be considered as having accepted it, if he keep the bill a great length
of time, or do any other act which gives credit to the bill, and induces the
holder not to protest it; or is intended as a surprise upon him, and to
induce him to consider the bill as accepted. Chit. on Bills, 227. When he
accepts it, it is his duty to pay it at maturity.