a BB6, or CV6 depending on what ship line you take, is the end-game ship you can get for a ship line. a BB6 is a gun ship, and a CV6 is a carrier class ship of which you use planes to do your bidding (fighters, bombers, scouts, all of the above, etc.).
Points also allow you to class your sailors into their nation and carrier (gunners, repairers, engineers, etc.), and you will most likely have an issue getting enough points to class your first crew on-time. To help with this, you can roll a second crew (stats don't matter much for this crew) and use them to gain points so that you aren't late in classing your starting crew (yellow crew) as these yellow sailors have good starting stats.
a fleet is a group of players that are able to participate in events such as Harbor Assaults and Fleet Wars that take place on a somewhat regular basis on each server.
if by premium you mean a premium account, it allows you to gain an extra 40% experience, credits, and points at the end of each game as long as you have stayed in the game for at least 5 minutes (actually it's 4 minutes and 30 seconds but the timer says 5 minutes). although it helps you level up faster, it is not necessary...