Smartphone touchscreen ribbon connector
FFC/FPC, like Oli says. There are common connectors, or ZIF "Zero Insertion Force" connector. In the latter you first insert the FFC, then click the top part in the bottom part to fix it. ZIF connectors may be handy, since otherwise the FFC may get crooked when inserting; in my experience the insertion force can be rather high. On one occasion we had our toolshop make a special tool for the insertion, when the connector was difficult to reach.  

The FFCs are often custom made, as their number of connections, pitch, and length may be different for each other application. Some have reinforced ends ("stiffeners") to cope with the connector's insertion force.