[chris (OP)]

When you see a helicopter at rest on the ground the blades look droopy and a bit flimsy. Obviously they cannot be because they are effectively supporting the mass of the aircraft when it is in the air. Once they spin up the blades also straighten, presumably owing to a centrifugal effect. What are they made of that gives them this incredible strength but also makes them floppy?

[alancalverd]

All wings flex to some extent, but fixed-wing passenger aircraft generally require fairly rigid wings to support the engines and fuel tanks when they are not flying, and small planes designed for rapid manouverability (fighters, aerobatic display planes...) need rigidity for their aerodynamics to remain predictable under high g loadings. The classic single-engine Cessna design uses a strut to support the high wing and also to transmit lift from the wing to the fuselage, so the wing, although very light, is effectively stiff and carries the weight burden of a strut. The equally classic Piper (low wing) design has the undercarriage fixed to the wing spar, so the inner wing is necessarily stiff by being heavier and more rigid.   

Gliders and helicopters aren't subject to the same constraints, so their wings and rotors can have progressively tapered stiffness: the tips only have to support themselves in flight and transmit some lift to the bit near the centre.

Centrifugal force (a curse on those miserable pedants who will say it doesn't exist - worthier lives depend on it!) is important in helicopter aerodynamics

As lift on the blades is increased (in a takeoff, for example), two major forces are acting at the same time—centrifugal force acting outward, and lift acting upward. The result of these two forces is that the blades assume a conical path instead of remaining in the plane perpendicular to the mast. This can be seen in any helicopter when it takes off; the rotor disk changes from flat to a slight cone shape.

If the rotor rpm is allowed to go too low (below the minimum power-on rotor rpm, for example), the centrifugal force becomes smaller and the coning angle becomes much larger. In other words, should the rpm decrease too much, at some point the rotor blades fold up with no chance of recovery.

Just one more reason why helicopter flying requires two brains and three hands.

[evan_au]

What are (helicopter blades) made of that gives them this incredible strength but also makes them floppy?

Helicopter blades need to have very high strength under the tension of centrifugal force.
But they do not need to be rigid, as centrifugal and lift forces make them lift up.

Kevlar and carbon fiber have these properties, and are popular today. A tougher abrasion-resistant coating may be used for protection of the leading edge and blade tip.
http://www.airliners.net/aviation-forums/tech_ops/read.main/46338/

It is amazing to see a dual-rotor helicopter like the Chinook power up, because the path of the drooping blades appears to intersect each other. 

https://en.wikipedia.org/wiki/Helicopter_rotor#Blade_design