Is this a Master Object - if so, then someone may have disabled the 'delete' button. To modify back, access the Object from ADMIN, access the 'Views', and layout for the 'List View'. You will see the 'Action Layout' at the bottom where the 'Delete' may have been pushed back to the 'Available' column.
Note, this will then be modified for all users, and it may then result in data integrity challenges if this data is used for look-ups on other attributes if you do delete the record.
This ONLY applies to Master Objects. If the Object is not a Master Object, the above will not work.