To add a little to Robert Knight's answer, since the title of the post itself doesn't mention foreign keys (and since his doesn't have complete code samples and since SO's comment code blocks don't show as well as the answers' code blocks), I'll add this for unique constraints. Either of these work to drop the constraint:
ALTER TABLE `table_name` DROP KEY `uc_name`;
or
ALTER TABLE `table_name` DROP INDEX `uc_name`;