The spider is a cobweb weaver - family Therididae (can be determined from the bulbous abdomen, thinness of legs and small cephalothorax) and appears to be Steatoda triangulosa from the brownish colour, white markings on the abdomen, and rings/annulations on the legs. S.triangulosa is found widely across North America and Europe.
The prey is not a caterpillar but a myriapod, I believe in the order of flat-backed millipedes.
As @DKNguyen already answered, the spider was preying on the millipede. The 'dancing around' was the spider trying to subdue the prey by wrapping it up in silk, but for prey that size, it takes a long time to get enough silk on it to hinder its movements.
Once the spider gets a meal out of the prey, they detach it from the web and let it fall to the ground below. This is often a way to detect when you have a cobweb spider in a corner or under a stair or something - when you see a clutter of dead bugs. The clutter also tells you that the spider is helping take care of pest control while staying in one spot.