Any idea of what the connector from the right is named?
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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.
Micro-MaTch Connectors Overview at TE