Applets: Java Applets are the cute little components that run within the confines of a web page. Unfortunately, their star has faded in the last decade, as newer client/browser side technologies such as Flash and JavaScripting have provided much of the functionality that was originally delivered by Applets. Nevertheless, there is still a place for Applets in many enterprise architectures that need slightly fatter clients on the browser side.
Servlets: An application server must be capable of handling and responding to requests that come in over the Internet. A Servlet is simply a Java based component that handles web-based requests.
The purpose of a Servlet is to accept and process a web-based request, and subsequently send a response back to the client. Since the client is likely a person surfing the Internet, the response delivered to the client usually takes the form of a web page.
Servlets are popular because they are incredibly easy to code. With only a little bit of a Java background, anyone can code a Servlet and dive head first into the empty pool know as server side development.
-Cameron McKenzie