PHP stands for PHP: Hypertext Preprocessor. It's a preprocessing spript language, which means that unlike JavaScript or something, it runs the code on the server side of the web, not the users. When using PHP the user cannot see the code written in the Source of the web page.
A PHP server is a server or basically a web hosting service that supports the use of PHP. If they do not support it, it will not read the PHP.
I am not sure about whether or not you can do PHP in something like HTML Kit, but I highly doubt it; I know the high end web editors like DreamWeaver can, though.
Check this site out, it has a lot of turtorials on web design: http://www.w3schools.com/php/default.asp