Please post a picture! I'd really like to see that.

If you don't easily find an answer, contact your local ag-extension office and (at least if you have a good picture or take in a sample) they should be able to provide you with an answer with no problem.

Around my neck of the woods we call this, Love Vine, Genus Cuscuta. Love Vine starts out from seed then wraps around the host plant and doesen't need it's root system anymore. Try and destroy as much as you can to prevent the seeds from coming up next year.
Norm

Thanks Maternut. Tomorrow I'm going to pick all the remaining tomatoes and get the plant out of there. Brandon7, I'll take a photo before I get rid of it and post it-probably tomorrow evening.

Tmenees,
Thanks, but I googled it after Norm produced a name. I've seen that stuff around here, but didn't know what it was.

Plant vampire! That is Dodder (love vine, strangle weed, devils guts) - evil stuff!!! Definitely commit yourself to destroying evey bit you can before it reseeds. I bought a load of topsoil contaminated with it about five years ago, I haven't seen it in three years now. Best of luck!