Creación de una parte del programa del lado del servidor

You can make code available through a servlet, callable by a URL, and run it on a tomcat server.

You start out by making a servlet. Create a class that extends HttpServlet.

To handle HTTP GET requests, override the doGet method:

@Override
public void doGet(HttpServletRequest request, HttpServletResponse response) throws IOException

To handle HTTP POST requests, override the doPost method:

public void doPost(HttpServletRequest request, HttpServletResponse response) throws IOException

You can make these methods execute logic, or retrieve data, and expose it as something like HTML, XML or JSON.

To map the servlet to the url, you need a web.xml archivo.

<?xml version="1.0" encoding="UTF-8"?>
<web-app version="2.4" xmlns="http://java.sun.com/xml/ns/j2ee" 
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" 
xsi:schemaLocation="http://java.sun.com/xml/ns/j2ee 
http://java.sun.com/xml/ns/j2ee/web-app_2_4.xsd">

<servlet>
    <servlet-name>nameOfYourServlet</servlet-name>
    <servlet-class>com.your.package.ServletImplementationClass</servlet-class>
</servlet>

<servlet-mapping>
    <servlet-name>nameOfYourServlet</servlet-name>
    <url-pattern>/api/my/servlet</url-pattern>
</servlet-mapping>
</web-app>

To deploy this to a tomcat server, you need your web.xml and compiled classes to live inside a WEB-INF folder. Zip the WEB-INF folder into an archive, change the extension of the archive to war, and drop it in your tomcat webapps folder.

I think this is about the quickest way to get server logic running, but your way of implementation depends on what you want to achieve, and what server you want to use etc.

It's probably worth your time looking into servlets a bit more. See este enlace.

Deberías usar ServerSocket y Socket and related classes. For more on this, check out the Tutorial de redes Java.