If you wish to use the Forum tool to facilitate individual or small group communication, such as for a student journaling activity or a small group project, you can use a Standard Forum (as noted above) to enable this.
You will need to repeat the steps below for each individual student, or each small group, so be sure to give each new forum that you create a name that will identify who will be working in it.
Start by creating a new Standard Forum. At the bottom of the page for Add a Forum, there is a tab to 'Restrict Access'. Click on this option to open up more options.
Restricting Access to a Single Student
Clicking on 'Restrict Access' will open up this dialogue box. Click on User Profile. This will allow you to restrict access to a single student.
Clicking on 'User Profile' will lead you to this set of options. Make sure that 'Email address' is selected under 'User Profile Field' and that the second field is 'contains', NOT 'equal to'. In the third space, you may enter the student's email username and click 'Save and return to course'.
Please note that this method requires that you make a Forum for each student.
Restricting Access to a Group
Another extremely helpful document to consult when setting up a forum using groupings is Using Groups with Moodle Forums. It offers a clear explanation of how the Common module setting options of "Separate Groups" vs "Visible Groups" affect the use of Moodle forums so you can choose the option that will work best for your purposes.