As someone else noted, there are two "changes" you can make: to the FILE NAME for your web page, or to the TITLE and content that appear in that file. Changes inside the file don't affect inbound links (with some exceptions); changing the file name definitely does "break links" that are out there.
Obviously, your best strategy is to NOT change file names, though you may change the content on existing pages, including using a "redirect" to send traffic to another page at your site (or elsewhere).
If you absolutely MUST change the file names, you should still maintain files with the "old" file names on your server; usually they will just "redirect" to the new versions of those pages. Do NOT just rely on the standard "page not found" error message.
The most common reason for "changing file names" is a decision to switch to a different scripting language; the original files might have been plain .HTM or .HTML files, but if you use ASP you'd usually need to create files with the .ASP or .ASPX suffix, likewise PHP files would have a .PHP or .PHP4 suffix; ColdFusion pages would have a .CFM suffix.
Note that it is possible to re-configure a dedicated server so that files with a .HTM or .HTML suffix are processed using the scripting language. I've done this for my server, which interprets all .HTM pages as .ASP pages, so I can include ASP scripting within my .HTM and .HTML pages. (This can ONLY be done if you have a dedicated server, not for traditional "shared servers," "virtual private servers," or basic "web hosting accounts.")
You should try hard to NEVER have any pages simply "disappear" from your site, but instead use the 301 or 302 redirect functions if a file is moved. However, it's always a good idea to also have a customized 404 error file that includes a short site map or redirect, so that visitors who do try to access an old page (or who mistype a page name) will see something besides a generic error message.