Working offline means you are working in your Windchill workspace without connection to Windchill. You will not be able to retrieve new files to your workspace or upload files to Windchill. When you reconnect, you will be able upload any changes that you have made while offline. Has little beniffit other than allowing you to work when connection to the server is not available.
No Server means you will be working in a file folder, on your computer or networked. All files are saved outside of windchill and can be time consuming to get back into windchill if you don't know the procedure to avoid file conflicts.
Working with current database files, working online in a workspace is the best senario. The data base is only changed when the file is checked in. Changes made are stored in your workspace and can be uploaded to windchill, outside of the common files, to prevent lose of data if your workspace is lost or corrupted.
Working with new parts only, using no server makes it easier to go back to a previous save for a file, as all iterations are in the working directory. However this usually means you have a higher risk of losing data and no access to common parts such as hardware.
There is always more to learn in Creo.
