Is it a good idea to bring your current companies business card to a new job interview?
No, because in a corporate setting you should only use your business card when you're representing your employer1 . Everyone who needs your contact details should already have them and they should be different from the ones on the card. You're not applying from your work e-mail are you? That said, according to Alison Green (#3 at the link) it's not really a big deal if you do hand one over.
When I was going for interviews fresh out of college, I already had personal cards made. Some of the interviewers I met gave me their card during the interview and promptly received my personal card in return. It's a bit of a gimmick, which I normally don't recommend when job searching, but all of them appreciated it. A few even mentioned that it added a professional touch. Personal cards are inexpensive and quite convenient to have on hand in general. They're also known as calling cards.
1 - It's fine to use your business card outside the office. As an example I routinely use mine when people have trouble entering my name in their records (and I've run out of personal cards).
No don't bring your business card, unless it is for a internal transfer.
The card is used to tell them the best way to reach you. You should provide that information on the resume and/or job application.
If you are talking to another company you don't want to use your business email and business phone number. Generally the point is to not let your boss know about your job search.