In Jira I have some existing Issues that are Stories. How can I place them under an Epic?
You can enable the Epic Link field to be shown (in your projects screen scheme). Then every card will have a field UI for selecting which Epic it is associated with.
This is the modern / current convention.
You can also configure the Board to group the cards by Epic in Configure Board->Swimlanes->Base Swimlanes on Epics. To really put issues "under" an Epic the Structure Plugin might be what you are looking for.
Under Jira GreenHopper (v6.0.6) there is a new feature. Jira > Agile > board > Plan, drag and drop the Story Issue onto the Epic.
Using an old view, navigate to the story page. There should be a Epic Link field. Clicking on it and changing the value of it, should move story to the appropriate Epic.