A Private Discussion can only be viewed by your Family Group, whereas a Public Discussion may be viewed by anyone on Geni.
To start a discussion, login and click on the Research tab to select 'Discussions'. Otherwise, you may simply go here. Then using the "Viewing" dropdown menu, select either "All Public" or "My Family" and click on "Start a New Discussion" to create a discussion.
Note: You may also create Public discussions on the profile pages of any public profile.