To compile servlets you need the class files for the Servlet API.
To run servlets you need some form of servlet container. This may either be a stand-alone servlet-capable web-server or an extension module which plugs in to an existing server.
The simplest way to get started is to download one of the complete, statnd-alone servers which implement the latest (2.2 at the moment) version of the Servlet API. This should provide you with everything you need.
The official reference platform for Servlet development is
Tomcat, from the Apache Jakarta project at
http://www.jakarta.apache.com/ . My favourite Servlet container is currently Resin from
http://www.caucho.com . Both are modern, high-quality, standards-compliant servlet containers which provide the current version of the Servlet and
JSP APIs. Both are free for most uses.