It would help to know what forum software and SQL/DB software you are using.
I am mostly familiar with phpBB, so I'll give you the rundown there...
You NEED access to the SQL server to make changes. You will need to be able to create a table and add, update, and delete rows from it.
You also NEED access via FTP or some other means to upload php scripts to within the forum directory, or otherwise some other location on the server.
In phpBB, moderator status can be gleaned by grabbing the users cookie, running a DB query for that user, checking the cookied hash (password check), and comparing the user_level field. If user level is 0, user has no priveleges. User level 1 is admin, User level 2 is admin.
Then you basically write a custom PHP script/form that does what you're talking about. Each report gets added as a new row in the new table. Each row has a few status boxes for various things. One of which could be admin_approved, which is default 0.
When admins log in to the script, they see a list of pending (admin_approved == 0) reports. They can choose to individually (or collectively) approve them.
Then you have another script that can run a check down the current reports, and if a certain user has a certain number of confirmed reports, they get whatever disciplinary actions. Then, you take all these reports and set a flag (action_taken or something) to 1 so they are not counted for future disciplinary actions.
The best way to do this would also have access to the forum software, where you can go add a button near each post so you can submit a report that way. Also, you could have it check the 'reports' tally against each user everytime they log in, saving you from having to manually apply the actions.