What are those clear, squiggly eye-floater things that sometimes get caught in your vision then move when you look around?

Floaters: It sounds as if you have had a posterior vitreous detachment. This is a situation where the vitreous(jelly) of the eye detaches (not a retinal detachment), it then "crumples" up leaving dots, spots, strands of blurry vitreous; i.e. The "floater". You should have an exam to ensure the incident did not tear the retina, which could lead to a retinal detachment. Can be normal strands and dots of vitr.

Vitreous Floaters: The back part of the eye is filled with clear jelly-like substance called vitreous. As it begins to liquefy over time, some of the fibers and debris become visible, particularly against a white background. New floaters and a change in your normal pattern of floaters can be a sign a of retinal damage or a pvd. You should get checked by an ophthalmologist.