UART adapter to... what is this connector's name?
The left one is called Micro-MaTch™ connector, and it's a board to flat cable 



or board-to-board connector. This is a two-part connector, but there are also versions where you solder the flat cable holding part directly, so that it can't be removed.



The other one is a board-to-board connector, like Jason says.  



Usually you'll have a couple of them, so that the boards can't rotate relative to the other one. This is tricky, though, as the least mispositioning will cause strain on the solder connections.

Further information
Micro-MaTch Connectors Overview at TE