Save:
'save' an object is done (by default) in your own department.
Share:
'share' an object with other users (persons or workgroups / who are in your own department or in another department), so they also get access to it, while the creator remains to have full control over it.
Assign:
'move’ an object to another department, so others will gain control over it as well, although the original creator still remains the owner of the record.
General info: user functionalities depend on the roles & permissions settings. If a certain functionality does not work within your user account, this will be because you don't have sufficient permissions for this.