A star rank is the next step up from the 'A' rank in 'Grand Prix'. To get a star rank, you need to get good times on all four tracks, and preferably finish in first position in all of the races, however if you do well in three of them, you might be able to get a second place on one track. Try your best not to crash or fall off the track.
You can get one, two or three star ranking. Three being the hardest to get.
If you get a star rank in all the cups, it will show online, same as if you get two or three stars in all the cups.