There are several possibilities for why you will want to rename a page (one that you have found or created):
The terms "rename" and "move" mean the same in this context. They just refer to different models for picturing the operation:
Since the system marks the page with the old name as new page, it applies the first of the two models. This model avoids "changing the history", a kind of "historical revisionism". An oddity is that the renaming itself, which can be considered an important event in the history of the page, is not recorded as such; it is recorded indirectly, namely in the history of the redirect page.
Note that you have to be logged in to rename a page this way; regarding the set user preferences:
With the correct page displayed, click on the "Move page" link in the sidebar. You'll be asked for a new name for the page, and given the option to also move the page's talk page. NOTE: Unless you know what you're doing, it's safest to say yes. Click the "move page" button and the page will be renamed to the new title. The old title will become a redirect page, so any links to the old title will still go to the new page. However, note that double redirects (pages that redirect to the original page), will not automatically follow to the new page, so you will have to refer them manually. (as explained below)
The "move page" function keeps the entire edit history of the page, before and after the move, in one place, as if the page were always named that way. So, it's preferable to use this method over just cutting text out of one page and pasting it into a new one; old revisions, notes, and attributions are harder to keep track of it you do that. (But you may have to if, for instance, you're splitting a page into multiple topics. If you do, please include a note in the new pages' edit summary and talk page stating where you took the text from.)
The move itself is shown in the edit history of the page with the old title. However, if the page is moved back, the edit history of the page with the previous title shows the latest move only, with the corresponding user name, but with the date and time of the first move(!). Accordingly, the name of the user who moved the page the first time and the date and time of the second move are lost (except that they appear in the recent changes).
If the new title already exists and isn't just a redirect to the old title, with no history, the wiki will tell you that you can't rename the page. You'll either have to manually merge the two pages together, or — only if there's no real content in the page — ask an Administrator to have it deleted or list it on your project's "votes for deletion" page in order to make room for moving the page.
Another thing to remember is that redirects to redirects aren't automatically followed (this prevents infinite loops and spaghetti linking). Always check the What links here for your page, and if there are multiple levels of redirects, go fix the links to point to the new location directly. But this can be troublesome because your new moving might be reverted soon. Take some time to make sure there is no objection to your moving.
Avoid moving a page while the edit box of the corresponding Talk page is open: when you hit "Save page" you overwrite the redirect to the new talk page (you do not get the usual warning that the page has been edited while your edit box was open) and get a duplication of the contents of the talk page, with your latest addition added to the old instead of the new one.
It is useful to copy the message "Page ... moved to ..." to the new talk page, especially if there has been discussion about the name of the page.
Normally, to undo a move from page A to page B, simply:
However, if page A has subsequently been edited, or the move software is behaving weirdly, only an Admin can sort things out:
"Move wars" are highly unproductive, and leave vast numbers of pointless redirects littering the place, which some poor soul will have to fix.
To swap pages A and B, including history:
See also: Help:Merge for information about merging page histories.