1) my search form's file name is "searchform.php", but in the Appearance > Editor > Template list it also has a name of "Search Form".
"searchform.php" is the name of the php file as it is viewed by the file system on your server. "Search Form" is a title that the Template Editor gives to the aforementioned file. Basically, "Search Form" is a human readable version of the filename.
2) And when you want to include it into somewhere such as the sidebar, you call it "search_form", as in <?php get_search_form(); ?>
get_search_form() is what is known a a Template Tag which is basically a php function that allows you display different parts of your blog in various different templates. This Template Tag looks for a file called searchform.php in your current theme's directory and if it finds it, it will include this file in the template. If the file is not found , it will print a default version of a search form to the screen.
Where do these alternative names come from?
I think they are just there to confuse new users:)
I've created a new newsletter subscription form called "subscribeform.php". I'd also like to give it a plain English name and include it in my sidebar.php. How do I do this?
Not sure about giving it a Plain English name, but it is rather easy to include into your sidebar. First, upload subscribeform.php to your server - into your current theme's directory. Next, open your sidebar.php file in a text editor and add the following code in the spot where you would like the form to appear:
include( TEMPLATEPATH . '/subscribeform.php' );
If your theme is widget enabled, you will want to put this bit of code outside of the "widget conditional"... it looks something like this:
if ( !function_exists('dynamic_sidebar') || !dynamic_sidebar() ) : ?>
Best of luck to ya!