What visa do USA citizens receive upon arriving in Thailand and how can they extend it?

Without a visa it is a 60 day visa exemption.

This can be extended with another 30 days at one of the Immigration offices for 1900 baht

If you haven’t bought a visa prior to coming to Thailand then you would get stamped in for 60 days on what’s called a visa exempt entry.

You can extend that once for 30 days at a local immigration office. There are over 80 around the country so you use the one that serves the area your TM30 (notice of staying that is filed by the hotel or house owner) is filed.