I define a class, and then I instate an object of that class type. I want to send this object to another Java application running on a different computer transparently. What is the best technology to accomplish this?
There are a lot of ways to do this. Here are some things to look into and you can pick the one that would work best for your application.
Object Serialization pushing the bits over a Socket
Pretty much any communication framework will allow you to push objects over a network in one way or another. You just need to review them and see which works for your application. A quick google should find even more methods.
A (de facto) standard to implement this would be to use a web service, for example using JAX-WSwhich is bundled in Java 6. See this tutorial for a java-first sample (i.e. using annotations). This is pretty straight forward and easy.
There are other approaches such as Serialization over a Socket, RMI, EJBs but, when working over the Internet, web services are a kind of natural choice as they rely on existing standards (SOAP, HTTP) and deal easily with firewalls (which might be a real issue for all other solutions).