What is this wire to board connector called?
The technology is called Insulation Displacing Connector Technology

An insulation-displacement contact (IDC), also known as insulation-piercing contact (IPC), is an electrical connector designed to be connected to the conductor(s) of an insulated cable by a connection process which forces a selectively sharpened blade or blades through the insulation, bypassing the need to strip the conductors of insulation before connecting.  

Source: https://en.wikipedia.org/wiki/Insulation-displacement_connector

See also: Molex about IDC

To further narrow down, measure the pitch, estimate AWG (it may be printed on the red cables), try looking for markings on the connector (some markings are molded with the connector having the same material/colour: changing the incidence of light may make it more clear)