If you are talking about the shape of an object...
A 'curl' is a twist or spiral in an object - usually for smaller and softer objects such as hair, fur, or fabric. A curl usually turns inward.
A 'crooked' object is something that is bent or misshapen. For instance a tree branch can be crooked. Crooked objects are often old or poorly made, for instance an old house can be crooked, as can the fingers of an old person.
A 'crook' usually means a criminal or untrustworthy person though. It is unusual to describe a specific misshapen object as being a 'crook.'
Hope that helps. :)