There are several options to communicate your Galileo to your PC. It depends on what resources you have available and how you plan to implement your code. You can use the Arduino IDE or program in C++, Python, or use node.js.
You might find useful the WebServer example of the Arduino IDE. You can look for it in File>Examples>Ethernet.
you'd need to sort out your constraints. E.G. is it just to send data from a target to a host? Or do you want to go via cloud? Which physical connection is available? Which communication protocols can be used? ...
Some high-level thoughts:
AllJoyn, or OIC might be a high level option - you would probably need to compile it for Galileo.
Lower level: MQTT for instance (or other protocols like OPC-UA, DDS, ...). You might be interested in the Intel Analysis cloud which supports REST and MQTT. There are samples for various languages on enableiot/iotkit-samples · GitHub (some examples also included in Intel(R) IoT devkit Eclipse IDE).
even lower level: sockets / websockets / serial [for instance in Intel(R) XDK for IoT you find a full blown Node.JS / HTML5 sample which reads temperature data on the target and displays it in a HTML5 Browser (or Android app for instance). In that sample websockets are used.
Thanks Intel_Alvardo,, I have tried this.
Thank you for your answer, I am currently researching AllJoyn and I will look into MQTT. I will post again once I have a viable solution. Your comments are much appreciated.