Callouts in SharePoint 2013 are now possible with minimal client side scripting thanks to the new callout.js framework. The fastest method is to embed the code into a code web part.
Useful MSDN starter topic: https://msdn.microsoft.com/en-us/library/office/dn135236%28v=office.15%29.aspx
In summary:
- Make sure to use SP.SOD.executeFunc to call your code when the callout framework has been loaded.
- Get an element as launchpoint from the document. If the element is dynamically created via javascript, a DOM get_element is necessary to correctly reference the launchpoint.
- Configure callout options, id and launchpoint are mandatory
- Create the callout
Check attached files for usage examples.