This is as opposed to continuation betting, which is a bet made in or out of position after you have made the last bet or raise on the previous round, and thus have the betting initiative.
The name "donk bet" derives from the common derogatory term "donkey" for bad players, because bad/clueless players will often make this play without knowing why they are doing it, simply because they don't know what's going on.
Under the right circumstances, however, it can be very good--by no means does donk betting necessitate bad play.
This is a donk bet because:
A) I was out of position (the button gets to act after me on all postflop betting rounds; therefore, he has position on me), and
B) I did not have the betting initiative (I merely called the button's raise on the previous round--I did not make the last bet or raise and thus I did not have the initiative.)
v. the act of making a donkbet
I donkbet the river with my weak hand, to avoid giving a free showdown.