It isn't all that complicated.
It started with straight billiards (also called the free game or libre) in which you need to make your cue ball (henceforth the white) hit the red ball and your opponent's cue ball (henceforth yellow). Players quickly developed technique that allowed them to trap the two balls in a corner or along a rail and make, essentially, an unlimited number of points. That is why there is no longer a world championship, although it is still played recreationally in Europe and elsewhere. And the Koreans have an interesting 4 ball variant that is frequently played on smaller 8x4 tables.
So back in the old days efforts were made to modify billiards to make scoring more difficult. Two different avenues were pursued.
Initially most popular, the balkline games were played the same as straight billiards. except chalk lines were drawn on the table and the rules prohibited making more than one or two (depending on the variant) points when the balls are in the same section of the table, as delimited by the chalk lines. Then you have to drive at least one of the balls out of the area. The lines are drawn 14 or 18 inches away from and parallel to the rails, making 9 regions. Where the lines are drawn, and how many points you can make with the balls in one area are denoted with numbers. E.g.,14.1 balkline has lines drawn 14 inches from the rails and you can only make one point with the balls in an area.
This is a beautiful game to watch. You can find a smattering of it on youtube but if you have a kozoom subscription there is loads of balkline you can watch. The beauty of the game is the way the player keeps complete control over all three balls. Because when they have to drive a ball out of balk they almost invariably send it down and back, or around the table, right back to the other 2 balls. The speed control fo the best players is incredible. Many, but not all, of the best players in 3C got their start in balkine. Ceulemans and Caudron are both past multiple world champions in the balkline game.
The other avenue in increasing the difficulty in making billiards was introducing a requirement that before the billiard is completed the cue ball has to have contacted at least one cushion - one-cushion billiards. This is also a beautiful game for similar reasons as balkline, the complete control the expert player exerts over all three balls. Lots of this on kozoom as well. Ceuleman and Caudron are also both past multiple world champions, I believe, at this. And since you do not have to draw lines on the table to play this you can try it yourself. Not so easy to average as little as 2.
Together, straight billiards, the various balkline games (whose numbers have changed from the old 14.1 and 18.2 because they are now only played where the metric system rules) and one-cushion are sometimes collectively referred to as the small games.
In the more esoteric realm, there is a 5 pin game that is played in Italy on a billiard table that is or was huge there. Big money involved. You can see some on youtube.
And there is some kind of Russian pyramid game that I don't know anything about .
