Do we apply for the free transit once we land in China? Also what does visa for third region mean?

If you transit in Beijing, you can apply for it directly after you land in at the airport and you needn't do anything in advance. If you transit in other cities, you need to tell the airline before you arrive and then apply for it at the airport.

Visa for third region means the entry permit for the next destination after you transit in PRC.

We are traveling to Singapore. Which we don't need a visa to enter. Will the third region entry permit still apply?

Angie, if you fly from USA, transit in a Chinese city where 72 hours free transit is implemented and then go to Singapore, then you can make use of the free transit in that Chinese city.