Flat profile IDC type connectors and cable replacement
What you're looking for to connect to one of those cables is a FFC or FPC connector.


http://uk.rs-online.com/web/c/connectors/pcb-connectors/fpc-connectors/
http://www.digikey.com/product-search/en/connectors-interconnects/ffc-fpc-flat-flexible-connectors/1442849

They come in both horizontal &amp; vertical orientations, and surface-mount and through-hole versions (although there seem to be more options in SMD than thru-hole).
When you're choosing one, pay careful attention to the pin pitch (spacing between adjacent pins) as this must match the cable you're planning to use with it.
Also be aware that there are top-contact versions and bottom-contact versions, so you'll need to decide which way you want the exposed ends of your cable to face (up or down) when you plug it in.
Finally, check the thickness of your cable so make sure that it will fit in the allowable range of the connector.