Clear liquid (vomit?) and worm-like object

Found a puddle of clear liquid (with some splatters around it) in my cat area, and there was a worm-looking thing in the center of the puddle. I thought maybe one of the cats had gotten a bone or something if someone had been careless with the trash, but when I touched it (I washed VIGOROUSLY afterwards) it was squishy/flexible and had little bits coming off of it.    What type of worm is this and what do I need to tell my vet on Monday to get them the quickest/best treatment? How much danger are my cats in?

Hello!  I can't see this picture as well as I'd like but wonder if this is a tapeworm segment eg Diplidium.  If you take the parasite or photo to the vets with you (perhaps in a sealed clear plastic bag if you take the specimen), they will identify it for you and make sure that you have an antiparasiticide appropriate to your pet. Worms are most frequently caught by eating mice etc.  Occasionally worms in the gut can cause an acute blockage, but it is rare.  Heavy burdens can cause tummy ache and other symptoms.  However the main short-term risk of parasites is usually a zoonotic risk to the owners / other animals, which you sound to be aware of.  Please do let us know how you get on.

The little 'bits' could have been eggs or larvae so you may want to consult a medical source or your vet to make sure that you are doing everything possible in order to protect yourself.