Flex vs stiff frames?

I've heard lots of folks say that their frame is very stiff, or very flexy. What does this mean?? Can anyone describe one vs the other? Are certain frame materials stiffer than others, or is it the geometry of the bike? Why would you prefer one over the other?

I'm just curious...I have no idea which describes my bike, lol.

If your chain rubs against the front derailleur when you are climbing out of the saddle or pushing hard on the pedals, that's a flexible frame. If your teeth rattle over bumps and you feel every frost crack or pebble on the road, that's a really stiff frame. You don't lose pedalling energy from flex with a stiff frame, but you give up a bit of comfort.
Bike manufacturers tend to make their race models stiff and their sportive type models a bit more flexible. Some designs aim to have vertical flex for comfort (the bike flexes upwards over rough surfaces with lateral stiffness to get the most efficient pedal stroke.