Elite is one of the best known space fighter/trader games. Anyone who has played privateer or space trader (on palm) will be very familiar with the style & genre of this game.
You start out with a weak ship with limited cargo hold inside a base near a sun. You are able to purchase goods or equipment. Money is made by purchasing goods at one location and selling them at another for profit. The price of the goods is determined by the system's stats, including race, technology level, resources etc.
In the beginning it is very difficult to work out what to buy and where to sell it. It is helpful to keep track of prices with a pad and pen, fly around a little and work out where you can make some money.
When you first launch from the base you'll notice it is a big grey rotating multi-sided spherical shape with a small orange rectangular docking port. One of the hardest things when you first start is landing. You can use the autopilot landing to get in safely, but this costs money which is quite precious at this point in the game. I would recommend spending some time launching and practicing landing before attempting to make trades.
You will not last very long on your first try. It would be a shame to spend an hour or so building up cash and equipment then end up crashing into the side of a base from lack of practice.
When you make your first jump to another system do not bother trying to fight any of the traders/pirates/bounty hunters/police on the way. Just keep running as fast as you can until you reach the base (a safe zone). Apparently the safe zone doesn't apply to you so any pirates that wander in here will be easy pickings for you (they will assume attack maneuvers but won't fire). Anyone else though and the police will have your hide.
As the game progresses, eventually you'll be strong and experienced enough to jump to the outer areas of known space. Here you may even meet aliens and collect alien artifacts from their destroyed ships. I think there is some storyline around a particular area but it is not necessary to follow or even find this for the game to be enjoyable (similar to privateer).