The connector is most likely a custom job, but it looks a bit like Samtec's Razor Beam series. Without knowing the pitch or physical dimensions it's difficult, if not impossible, to find the exact part (even if it is off-the-shelf).
Regardless of what the actual manufacturer and model number is, this type of connector is used for high-speed connections between boards, where contact resistance and impedance are important factors in how the signal is transferred between the boards. Regular pin headers tend to have higher contact resistance and uncontrolled impedance, which can cause discontinuities in the signal paths. These types of connectors shown in your photo, as well as Samtec's Razor Beam series (among others) are generally designed with controlled impedance to minimize discontinuities and improve signal quality.