First, you must check that htaccess overrides are permitted in your server space - they usually are but may not be. If you can use htaccess with your site then either create an htaccess file in your site root (if one does not already exist else add to the existing file) and add these Apache directives:
RewriteEngine on
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}\.php -f
RewriteRule ^(.*)$ $1.php
Replace php with html if you are removing the html file extension.
If you already have pages indexed in search engines or linked to from external web sites, replace the last rewrite rule with this:
RewriteRule (.*)\.php $1 [R=301,L]
This will ensure that any incoming links pointing to the page+file extension will be correctly redirected to the page without the file extension and will inform browsers and search engines that the URI has changed.
Note - don't remove the actual file extension from your file names. Files are files, URI's are URI's and your files still need the extension even when this doesn't display in the URI.