Confi and Super-Confi were created by George Rosenkrantz, and are designed to facilitate slam bidding opposite a balanced hand.
'Con' stands for controls and 'fi' stands for fit. The general idea is that one bid asks opener for number of controls, and if responder gets an encouraging answer, the partnership looks for fits. Both players show suits up the line (at least Qxxx) until a playable fit is located. (There are ways to determine whether the jack is present, but I don't remember how right now). The convention is used when responder's hand is balanced (4432/4333) or quasi-balanced (5332, 5422). Confi is used when a small slam is in the picture; Super-confi when a grand slam is possible. Confi stops if the partnership as 9 or fewer controls; Super-confi if the partnership has 11 or fewer controls, counting an Ace as 2 and a King as 1.
For example, assuming a 15-17 1nt opener with 2s as confi, opener would bid 2nt with 3 or fewer controls, 3c=4, 3d=5, 3h=6, 3s=7, 3nt=8 (some would collapse these responses and show 4 or fewer, then 5, etc). If opener's contols plus responder's controls total 10 or more, then they start bidding suits up the line.
Similarly, if 4d is superconfi, then after 1nt-4d-?, opener would bid 4h=3 or fewer, 4s=4, 4nt=5, 5c=6, 5d=7, 5h=8. If the total number of controls is not 12, responder signs off at 6nt. Any other bid initiates a search for grand slam strain and promises all 12 controls
---- I received this explanation from Henry Sun on discuss-OkBridge after requesting info on the above conventions. From: Henry Sun

As an additional note, others on the discussion group confirmed that 2S was the standard CONFI asking bid, and 4D was the standard Super-CONFI bid.