Cards are pieces of evidence in your case. Fun fact: Evidence in debate cases is called cards because in the time before computers, people would cut of evidence from a magazine or a scientific journal and put it on a 3 by 5 index card. Cards usually must come from a reputable academic source, a reputable news publication, a government agency or others that have a similar reputation. Cards in a speech are usually introduced in a speech with something like "According to _," Cards can also be identified by using the last name of the main author and the last two digits of the year the publication was published, for example, "Can I see your Johnson '13 card?"
Dropping refers to the act of not adressing an opponent's point in your rebuttal. If your opponent calls you out, this can usually result in a loss, so don't do it.
A field that encompasses Debate, Public Speaking, Extemporaneous Speaking and Interp/Speech.