Moving (renaming) a page means giving it another name. This is done by using the "Move" tab at the top of the page. The tab is not visible if you are not logged in or do not have permission to move pages.
If you move page "A" to a new title "B", this operation will do the following:
The second point is important. By using the "Move" feature, you also move the editing history, which is desirable in most situations. The editing history allows people to see how the contents of the page were built up, and who contributed what. If you move contents around by copying and pasting (not using the move feature), you do not achieve this.
Because the move operation leaves behind a redirect (see Help:Redirects), it should not result in any broken links, however you might want to hunt down pages or other redirects which link in to the redirect, and change their text to link directly to the new title. The "Links" feature, which can be used through the tab at the top of each page, is useful when doing this.
As with all wiki editing, a move operation can be reversed (by any user). To reverse a move operation, simply move the page back, e.g. move page "B" back to "A".
The page title "B" will still be left behind, as a redirect from "B" to "A". Any user can remove the redirect and propose that the page be deleted. However the final deletion will require sysop privileges (see Help:Deleting a page).