Web site "Designer" is an often misued term. The designer may or may not know anything about programming modern websites besides the client side interface of the website using CSS styles and Javascript in a WYSIWYG editor. A web Designer may also know how to code AJAX interfaces and bind it to a Web Service.
Web Developer is acutally has to know how to program a server side scripting language like PHP or JSP and a database resource like T-SQL. They may also know how to create a simple or rich API for the site or merely a REST or SOAP web service.
PHP and JSP are server side application scripting languages for Web Servers. Web developers code the website backend using a server side scripting engine (like PHP and JSP), and it streams just basic HTML (and Javascript) to the client viewing the website.
Essentially a client Web browser can only process HTML code, Javascript code and some CSS code, nothing more and this was usually enough in the 1990's because websites were simple. Web developers wanted to do more with websites, but were limited by the much too basic featureset of HTML on the client side, so they created a plugin framework that would extend functionality to traditional Webhosts - this would come Server Side Scripting.
The client requests to the webhost server were still all done in HTML (so no new browser was needed on the client side), so the web devs programed the basic websites that clients would view with special instructions that were only run on the Webhost via the Scripting Extension (like PHP).
Now with a single button click on from the client, the webhost could execute a extention script that could do math, query an SQL database, access other websites behind the scenes etc... and once completed, the webhost would return the results to the client in just HTML, which was viewable with any web browser.
All Web 2.0 websites use this model and not static HTML like from the 90's.
Avoid JSP - it is a Java Servlet and runs in the Java VM, so performance is terrible on even a small scale website.
PHP is probably the most common and widely used and there are plenty of free resources and tools available (like LAMP/WAMP/XAMPP) for learning.
Other languages that real application developers might enjoy are Django (Python), Ruby and Perl.
