What type of connection is this?
It's an FFC (Flexible Flat Cable) cable. Similar to FPC (Flexible Printed Circuit) which it is often mentioned with. 

You can buy the necessary connectors from places like Mouser, Farnell, etc. Here is the selection from Farnell. 

If you can't find any datasheet or details, you will have to count the connections and measure pitch (distance between each copper pad) to figure out what connector you require. If you provide the info (close up photo against ruler would help too) we can help you select the right one.