What are types of connection between metallic conductors and glass part of an LCD?
The conductors one the glass cannot be soldered to, so a few methods are used. 

On big crude displays, metal spring pins are glomped onto the edge of the glass and secured with (usually clear) epoxy. 

On smaller displays, elastomeric (Zebra) connectors are often used. 

Another method (suitable for the highest densities) is to heat seal an FPC (flex circuit) to the glass with conductive adhesive. Sometimes the FPC contains active components, other times it's just a jumper.  

It's not at all practical to remove this from the glass, or replace them for a one-off, you should replace the entire display- usually the other end of the FPC goes into a socket. If it's heat sealed onto a PCB that's going to be more difficult, but that would be the end to be fiddled with. 

If you had the exact replacement FPC (not likely), it might be possible to fiddle around with a well controlled hot air gun and re-laminate the exact replacement cable with the appropriate equipment creating the proper temperature and pressure to get a reliable bond, but that seems quite unlikely to turn out well without proper equipment and jigs for the particular display.