You will move a page when you want to organize differently your pages.
All you have to do is to edit the pagen and use the Parent control to point to a new parent. And you're done.
See screenshots and the description of the edit form available in the Edit page.
The parent relationship is decided by pages written in the site's main lang. The so called master lang. For example, on this site this is english.
Only those pages written in the master lang can be moved. Other pages, the translations, are linked to the master page they translate. They will be moved along automatically when their master page is moved.
That's right !
The thing is how do you move the master page ? The Edit form shows you existing pages in the Parent control. What would be the point of making the root page be the child of another page which is directly or indirectly a child of the root page ?
What could be done is swap them, well yes, that might be possible but this is not coded yet.
Instead there is another thing which is available : any page can be made the new root node using the 'Make root' link available in the edit area.
WARNING : this is available only for users having moderator privileges !
In that case, here what will happen :
That's complex isn't it ? Well quite. See it this way, the new root page is elected as root page, the old root page is attached to the new root page. Children of the new root page do not follow it, they are attached to their grand father instead.
I don't know if it's the best solution, tell me if you have a better one. It should not be that difficult to implement. Anyway, changing of root page should not happen that often...