You take off and go into hyperspace. You get into the next star system and fly to a space-port. You get attacked by pirates, the military, unhappy maffiosi, or other armed brigands. You kill them all, and land at the port for repairs and trading. Combat is improved from Frontier, but not the same as Elite's.
Some people find drifting in space quite confusing, and have a hard time dog-fighting.
In FFE, enemies attack in swoops toward you, and never stay flying away from you as they did in Frontier. They usually attack in ones, but quite frequently attack in groups, in which case it can get quite dangerous. I had no trouble killing off the ravening hordes, but I had loads of practice in Frontier. The special missions can be very entertaining. I accepted an extremely well payed one in the first couple of months of gameplay, and assassinated the leader of a faction in a civil war which had caught the interest of both the imperials and the federals.