Here's how 3 bottles are typically used:
Bottle 1: Preliminary Round - Used when the number of entries in a category (or combined category) is greater than around 10. That is the most a panel usually wants to judge to guard against palate fatigue and to keep the time for each flight to a reasonable amount. When multiple flights in a category need to be judged, then the top entries from each are passed on to a final round to decide the winners. E.g., a category has 20 entries. Two flights of 10 are judged in a preliminary round, and the top 3 to 4 beers from each are passed on to a final round.
Bottle 2: Final Round - Entries passed from the preliminary round are evaluated in a single flight to pick the winners (E.g., 1st, 2nd and 3rd.) You don't want to select winners just based on score from the prelim round as there may be skew between judging panels. Judging them with a new panel in a single flight ensures each entry is evaluated in relation to the other top entries in that category. Having a bottle for prelims and another for a final round means the judging doesn't have to be done same day. (Judging a flight can take a few hours and judges are volunteering their time often on weekdays.) Smaller comps (< ~100 entries) don't require a preliminary round and just do final judging on each flight. That's why sometimes one less bottle is needed.
Bottle 3: Best of Show (BOS) - Once the winners from each category are determined, the 1st place winner in each category go on to a BOS round where a single flight is evaluated (usually not scored) for the best overall beers in the whole comp. Some comps request 3 bottles, but will allow you to send only 2 bottles but then take you out of BOS contention. Read the rules when in doubt. If it isn't explicit, send 3 or risk disqualification.
An interesting (and perhaps controversial) comp is NHC where the prelim / regional rounds only require 1 bottle, yet the number of entries is still large. (E.g., West had over 600 entries.) Beers in a category are judged by multiple panels and the top beers from each panel are passed on to a BOS-type round. It's called a mini-BOS in this context, but the concept is the same. The 'odd' thing is that a single beer is used for both the prelims and mini-BOS to limit the logistical issues of storing and transporting so many entries. The disadvantage is a beer opened early in a flight will loose some carb and aroma (even if capped) while waiting for the mini-BOS to occur. The time between the prelim and final mini-BOS judging is minimized by using something called queued judging, but that's a different thread. ;-)