What does 1080 i refer to?

1080 vertical lines of resolution, but the i stands for interlaced, as opposed to p which stands for progressive.
The difference is that interlaced updates odd lines, then even lines alternately. Progressive updates the entire image with each scan. If your TV set can display 1080i then it can also display 1080p (which transmits all the vertical lines at once) although the only media capable of a 1080p signal is a Blu-Ray disk.