Moving a page renames it and moves all of its history to the new name. The old title will become a redirect page to the new title. You can update redirects that point to the original title automatically; if you choose not to, be sure to check for double or broken redirects. You are responsible for making sure that links continue to point where they are supposed to go.
The page will not be moved if there is already a page at the new title, unless the latter is a redirect and has no past edit history. This means that you can rename a page back to where it was renamed from if you make a mistake, and you cannot overwrite an existing page.
Note: This can be a drastic and unexpected change for a popular page; please be sure you understand the consequences of this before proceeding.