Giving Add and remove option to select and deselect from one list to another I want to make user able to chose from one list of select options to another . A good example is in the picture here .

. Could you please tell me if there is a module to achieve this? 
I tried using two select elements in the form and some jquery . But that is not working perfectly . 
The problem is when user once select the options and visit next time , the previously chosen list does not appear . Thank you .
I think you're probably looking for the Improved Multi Select module.

The Improved Multi Select module replace the default multi-select boxes with two pannel list and search.
The first panel holds the options that are available, and the second panel holds the options that have been selected.
Two panels are separated by &quot;add&quot; and &quot;remove&quot; buttons. You can select an item from the first box, click the &quot;add&quot; button, and add it to the second box. Likewise, you can select an item from the second box, click the &quot;remove&quot; button, and it goes back into the unselected box.

The screenshot from the module page looks pretty similar to the screenshot in your question, except there's no drag-and-drop re-ordering built in (you would need to implement that yourself I think):