When you have a FX position you are going long one currency and short another. The interest cost difference between the currency you are long and short is called the carry.
If the interest received on the currency you are long is greater than the interest you are short. It is called positive carry. Positive carry trades are put on with the earning a positive yield from owning high yield currencies like the Australian dollar through funding the trade by shorting low yield hence cost currencies like the Japanese Yen.
Negative carry occurs when there is a cost of holding a position when the interest cost the currency sold is higher than the interest you receive on the long position.