The verb consign means to transfer permanently to another. You can consign ownership of your old car to your son, an act that will probably make you the "best parent ever" in his eyes.
Consign means to commit or relegate. Those platform shoes you still have from high school in the 1970s? You should probably consign them to the trash — of course, they probably should have been consigned in the 1970s! Consign can also be used if you turn over an object for sale where the business making the sale gets a percentage of the profits and so do you. If you don't like the art work you inherited, you might consign it with an auction house and use the profits to buy something you like better.