Thank you for your question, and thanks for using JustAnswer.com.
Your cousin can give you a car, that's completely acceptable. Your cousin would fill out a form 709 for the IRS if the fair market value of the car is over $14,000. If it's less than $14,000 there's no reporting requirement. Your cousin will owe no gift tax because you have a lifetime exclusion of up to $5 million in gifts.
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You should be aware that your question, as with most tax questions, can never really be answered completely...addressing all the permutations; that is because of the many assumptions that have to be made I have done my best to determine what I think you are asking and answering it in the most direct and understandable manner possible. If, however, after reviewing the questions, you have any uncertainties or further questions, please do not hesitate to ask.