Read the WordPress Codex on the subject of Roles
http://codex.wordpress.org/Roles_and_Capabilities
To confirm that the system works the way you want, as an admin create a test member with the author role and sample content as you would for a ‘real’ member. Then sign out and sign in using your test member’s credentials and see what capability that test person has vs. what you want them to have.
The Author or Contributor role should do what you describe. Otherwise you can always generate a custom role cor your site which has the capabilities that you want and assign your members the custom role.