Who went to jail, if anyone? Was there any new laws introduced because of this? How did they get found out?
Free agency was not introduced to baseball until the 1970's. Before that, a team more or less owned a player's services. Once a player was with a team, he could not go to another team by choice. This meant that there was little bargaining for salary. A team would determine a player's salary and the player would have to accept or not play in the league at all. This imbalance of bargaining power meant that there was no free market for players. Charles Comiskey, owner of the Chicago White Social was notoriously stingy. Many White Social players felt that they were underpaid and abused.
Baseball was not an extremely profitable career for players like it is now. Many players had other jobs in the off-season. This meant that gambling was not uncommon in baseball, but until 1919, there were no allegations of fixing a World Series. The White Social were heavy favorites over the Reds so a group of gamblers figures that they could get a good payout if the Reds won. Knowing that the players were upset and could use some extra money, the gamblers came to a couple of White Sox players, who spread the idea around to several teammates.
The Reds won the World Series and 8 White Social players were accused of fixing the games. Those 8 players were banned from baseball for life. Up to that point, baseball was run by the President of the American League and the President of the National League. After the Black Sox scandal, the league wanted to come down hard on gambling and appointed Judge Kennesaw Mountain Landis as the league's first commissioner. He was certainly good at getting gambling and other undesirable behaviors out of baseball. However, Comissioner Landis was a virulent racist and vehemently opposed integration of baseball. He served as Commisoner until his death in 1945. Two years later, Jackie Robinson debuted in the majors, but had someone else been on charge, racial integration in baseball would likely have come sooner.
Eight Men Out is a terrific book on the scandal and a wonderful source if you want more comprehensive information. The movie based off the book and starring Charlie Sheen is also interesting and informative, but it has a bit of the Hollywood treatment.
Edit: viral to virulent
Just to add this, they were indicted in Sept 1920 and the trial began in June 1921. The 8 Sox were suspended by Comiskey in Sept 1920 when they had like 3 games left in the season and they still had a chance to win the pennant. The Sox were charged with three charges including something like committing a public fraud; it was an odd set of charges as I recall.
They were acquitted! It was apparent the jury was friendly to them, I guess they were shaking hand with the jury at the end. Comiskey being an a-hole didnt hurt either; they were seen as the underdogs at least by the jury.
It didnt matter though, a day later: Aug 3, 1921 Landis issued his decree banning the 8 of them permanently. So basically no one went to jail.
As to how they were found out, I really cant remember the details. There were rumours about this even before the series began with Christy Mathewson, famous ex player, and acting as a reporter began marking down any plays he thought were suspicious. I guess there was some sort of gamblers or someone who tipped off Comiskey but the rumors were there from the start.
There's still alot of this that remains a mystery and probably always will. Joe Jackson's role is probably the most disputed; he didnt attend any meetings, but his name was tossed around by his teammates to the fixers. Did he really get his share of the money? And how did Gandil make off with $35,000 of the money (they were supposed to get $10k apiece). Gandil said later they gave up their attempt to fix the series but he's hard to believe. Jackson is sort of believable because he didnt make any really questionable plays in the field and got his share of hits, although none of them in key moments. Its really confusing what exactly was going on.
I highly recommend the landmark book on this subject Eight Men Out (the one the movie is based on). It's a great read and will teach you everything you ever wanted to know about this issue and more.
The basic gist is that the White Sox were the best team in the league but were paid the least in the league. They were teased about it by other teams' players and all-in-all did not feel good about it. This was well before baseball had the strong pro-player Union and contracts that it has today, so there was quite little that could be done about it "through proper channels."
Gamblers took advantage of this. By offering the White Sox players a large amount of money (much of which was never delivered on) to throw the games in a convincing way. The Reds, their opponents, were heavy underdogs, so gamblers who knew to bet on the Reds got insane payouts.
Eventually people started to catch on (at first, reporters were outright lambasted when they suggested that certain errors made by Sox players looked fishy and that the game might be thrown, but eventually evidence started to be uncovered and players started admitting to it.)
The entire story is a great drama, so again I recommend reading up on it some. It's an event that was perpetrated by big time gamblers at least as much as the players themselves, but the players got the punishment in the end.