Cap: for smaller containers
Cover: to hide the thing
Lid: to top or cover a container
A cap is used for smaller containers such as a toothpaste. It may be used on a bottle or a tube. A lid goes on jars and other containers.
A cover could be considered larger than a lid, but not always. Also, lid is usually only used when it is a cover on the top of something. Also, a lid can only ever cover a hole in something while a cover can cover a hole and cover the whole object as well.
To make things even more difficult, there are some occasions when only one or the other is correct. For instance, a teapot has a lid; a box has a lid; a dustbin has a lid; a drain can have a lid or a cover, but a manhole only has a cover!