You get a boarding number based upon when you checked in for the flight. They board in groups of 60, "A" group goes first, "B", then "C". Recently (within the last 5 years or so) they've reserved the first 15 or so "A" slots for their "business class" and for people who pay an extra $10 to get priority boarding. There isn't any fighting because everyone is assigned a certain spot in line. In a practical sense, you'll get your choice of seat (aisle, window, etc.) if you are anywhere in the "A" group, and out to perhaps B30 or so. If you're a "C", you got the middle seat.
For the way that I used to travel, which was almost always by myself, I found I preferred their seating process over other airlines. On other airlines you basically also get boarding priority based upon when you check in, but you then are only given a choice of the seats they offer you. Lately airlines have been withholding exit aisle and other seats with a little more leg room and then offering that if you pay them $25 more, you could get those seats. Southwest doesn't withhold any seats, so if you're traveling by yourself, you still have a shot of a random non-middle seat even if you are way back in "C".
On Southwest, back in the day, so to speak, you used to get your boarding number in the order you arrived at the gate, so if you came in on a connecting flight that was late, you were going to be in the back for the next flight. Relevant to this story, if you are traveling with a child under 4 years old, you are allowed to board during "family boarding" which is just after the A group boards. My guess as to what happened here is that he got priority boarding based upon his frequent flying status, and his kids probably had "B" or "C" tickets, so he tried to board them with him (and if he really flies that much with Southwest, he would know he's not allowed to do that); he would have had the option to "upgrade" their tickets by paying an additional $10 each if he wanted to get them up in the early "A" section. I think he figured he didn't want to pay the extra money to upgrade his kids and that he could slip them into boarding with himself counting on the fact that either he felt entitled due to his frequent flying status, or that the gate person wouldn't call him out on it.
Do families get to pre board?
An adult traveling with a child four years old or younger may board during Family Boarding, which occurs after the "A" group has boarded and before the "B" group begins boarding. However, those Customers holding an "A" boarding pass should still board with the "A" boarding group.
Can groups assigned to different boarding positions board together?
Yes. However, in order to maintain the integrity of the boarding process, we ask that earlier boarding positions board with the later positions. For example, if a passenger is assigned position A16 and wants to board with a passenger assigned position A45, the passenger holding the A16 boarding pass should board with the A45 passenger.
Top Ten Things Overheard At The ANSI C Draft Committee Meetings: (5) All right, who's the wiseguy who stuck this trigraph stuff in here?