"Tournament Rules" are slightly different from the original, standard rules of Hearts at World of Card Games:
These rules are called "Tournament Rules" because they were used in Hearts tournaments that were run by Joe Andrews some years ago. The idea is to place less emphasis on luck and more on skill in the game.
If you have no strong preference between the original "Standard Rules" and the new "Tournament Rules", then you should set the Option for "Tournament Rules" to "Either". When joining an existing Hearts table, you may be seated at either type of table. The "Either" Option will get you seated most quickly.
If you like "Tournament Rules" and prefer not to play at tables that use the original "Standard Rules", then you should set the Option for "Tournament Rules" to "Yes". This can lead to longer wait times, because other people may be playing at tables using the "Standard Rules", and you will not be seated there. You will be able to see these tables in the "list of tables", however.
If you like the original "Standard Rules" and prefer not to play at tables that use the new "Tournament Rules", then you should set the Option for "Tournament Rules" to "No". This may lead to longer wait times, as well.
When you click the big "Hearts" game panel image on the home page, you will normally be seated at an existing Hearts table that satisfies all your Hearts Options. Sometimes, no such table exists, and a new table must be created for you. If you want any such new tables to use the "Tournament Rules" Option, you must set the Option for "Tournament Rules" to "Yes". If the "Either" Option is selected, you may join either type of table, but when a table must be created, it will be one with that uses the "Standard Rules".
You can always change your Options, so don't be afraid to experiment by choosing one Option or another. You can switch back and forth as long as you are on the home page.
If you use the list of tables to look for a table, you will be able to distinguish those tables with "Tournament Rules" from those using "Standard Rules". The "Tournament Rules" are listed with a "tr" next to them. Take a look at the screenshot below. The first table in the list uses "Tournament Rules" (tr), the second does not, but uses the "Allow First Turn Hearts" Option, and the third is just a regular "Standard Rules" table: