Premium income has related, but different, meanings in derivative trading and insurance. In the context of derivatives it means the amount received by an option writer for writing an option. In the context of insurance it means the revenue a company receives as premiums from its customers.
The similarity between these can be seen, if one regards an option as providing insurance against an adverse price change.
Both these numbers tell you how much revenue an insurance company actually made from its customers: how big the core business is.
The complexity of insurance means that it is necessary to look at this number in conjunction with many others. The present value of net new business premiums and annual premium equivalent are also important indicators of growth (as well as change in premium income) and, particularly for life, investment performance is as important as sales.