It is an Echeveria, which is a succulent but not a cactus. E. runyonii, perhaps. (E. runyonii is different from E. runyonii 'Topsy Turvy').

Is it the flowering stem that was snapped off?
What's left - can you post photo?
If anything left (cat didn't ruin/chewed up the roots/rosette), it should live.

Rina

This is what's left. The roots are fine, the most beautiful piece (in my opinion) broke off right from the middle. I was worried its not going to be able to produce another flowering stem quite like the one that snapped off. Thank you all so much for your help. I appreciate it more than you know.

Looks like you still have 'mother' plant, flowering stem is broken off.
The main plant should continue growing. Just make sure it is potted in very well draining mix, and receives enough light-sun. It will eventually bloom again.
You can try an experiment: stick broken off piece in the potting mix, it may develop some roots. You have nothing to loose - if it doesn't. Flower buds will probably not open, but you may have a new plant.
Also make sure that you don't overwater the plant. What zone/location are you in? Did you keep it inside or outside?
I am not sure about ID-hoovb already suggested possibility.
Rina

Cut off the flowering stem. Do a clean straight cut on the bottom of the broken stem--just get it flat. Let the cut dry for couple or three days in full shade. Once the cut is dry, place the stem into a shallow small pot of DRY gritty mix. DON'T WATER IT. Put it into bright, indirect light. It will root. Check it every once in a while. Once you see some nice roots forming, put it back into the gritty mix and water it sparingly for the winter. You can mist it lightly on a regular basis. Increase watering when spring arrives.