PHP = PHP Hypertext Preprocessor
PHP is a language. There are many, many languages that can accomplish the same task.
OOP = Object Orient Programming. While not an all-encompassing definition, you can think of it very basically as the ability for the programmer to create separate, reusable objects.
You need a server to test the code because PHP isn't compiled and is a server-side scripting language. The server with php installed goes through your file and executes the instructions, then passes the results back through your browser. You can use something like Zend Studio (http://www.zend.com) to debug your code if you want to test locally and don't want to install PHP.
HTML is a markup language. It doesn't actually do anything. One way to think of it is that HTML simply spits out formatted text. Scripting languages like PHP let us use things like variables and control structures to perform conditional actions. For example, we can tell php to print out "hello" if x condition is true, and print out "goodbye" if x condition is false.
Hope that helped.
