What are rubbery, conductive strips called on the sides of an LCD screen?

Often called Zebra strips (or Elastomeric connectors). They have very thin vertical conductors that connect between flat PCB pads and things like LCD pads on glass.

Here is a link with a similar pink component.

http://en.wikipedia.org/wiki/Elastomeric_connector

As others note it's a "zebra connector" which conducts through the plane of the rubber but not across it (ie in the directions you'd expect for it to work).

The 'pitch' of the band s of conduction/non conduction are finer than the pitch of the contacts concerned so that short circuits are not formed between the contacts when the strip is moved "sideways".

Use an Ohm meter to measure resistance through the rubber between directly opposite points and then between points which are offset.
You will find that

• When the probes are opposite each other there will be a low resistance path between them.

• When the probes are offset by more than the thickness of the conductive bands that the resistance will be very high (or more :-) ).

This method allows ease of assembly, self alignment of the "connecter" - as long as the LCD contacts and PCB contacts are aligned the rubber strip alignment is non-crucial.

Zebra strip inter-stripe pitch too large - shorting can occur

█ = contact
▓ = zebra strip conductor
░ = zebra strip insulator

Alignment OK:

████  ████  ████  ████     
▓▓▓▓░░▓▓▓▓░░▓▓▓▓░░▓▓▓▓░  
████  ████  ████  ████    

misaligned - shorts occur

████  ████  ████  ████      
▓░░▓▓▓▓░░▓▓▓▓░░▓▓▓▓░░▓▓▓  
████  ████  ████  ████    

Zebra strip inter-stripe pitch small wrt conductor pitch - shorting can not occur

Alignment OK:

████  ████  ████  ████  
▓░▓░▓░▓░▓░▓░▓░▓░▓░▓░▓░▓    
████  ████  ████  ████    

misaligned - shorting still cannot occur

████  ████  ████  ████
░▓░▓░▓░▓░▓░▓░▓░▓░▓░▓░▓░▓    
████  ████  ████  ████

See also this answer