There are two basic kinds of star system on the galaxy map: "Explored" and "Unexplored". For "Explored" systems, you can see the entire star system on the system map; there is nothing else to see or discover in them, for the most part.
"Unexplored" systems, whether they are truly "unexplored" or not, still earn you money for exploring and scanning them. You only get the 50% bonus if you truly are "the first", but by now, there pretty much aren't any "firsts" within inhabited space. But exploration data is still worth acquiring and selling; Universal Cartographics pay just as much money to the 10,000th person to scan a system as they pay to the second person to scan it.
For "Unexplored" systems you have never, ever visited, you cannot access the system map at all. To see those system maps, you can either (a) visit there, with a discovery scanner of some kind fitted, or (b) purchase the map data from the Galaxy Map. Purchasing the data is only available for some systems, but after you purchase data, the system becomes just like an "explored" system: all the stars and planets are visible to you. Think of it as if a company selling car GPS / satnav units has all the big cities installed in the unit for free, but the little country towns need a paid map upgrade.
When you visit an Unexplored system for the first time, you automatically detect the primary star. Detection is what the game calls a "Level 1 scan". Looking at the system map now, all you will see is the star you just auto-scanned and any space stations in the system, which will appear to be hovering in deep space orbiting nothing. They aren't really, of course, they're orbiting planets - you just haven't detected the planets yet. To detect other planets in the system, you need to actually activate and scan with your discovery scanner. Switch over to your Fire Control Groups panel, and assign the Discovery Scanner to a fire button - personally, I always assign it to the No. 2 weapon, since I hardly ever use my #2 for anything else. Once the scanner is assigned, switch your fire controls so that the scanner is ready to "fire", and fire it. A blue charging bar scrolls across, it takes about 5 seconds. You'll then hear the "foghorn" sound of the scan completing - this is known as "honking the horn" in explorer parlance.
If you've got a Basic scanner, your "honk" just discovered every star, planet, moon and asteroid belt within a 500Ls radius of your current position. An Intermediate scanner will detect out to 1000Ls, and the Advanced is well worth the hefty pricetag as it detects every single thing in the system; no more worrying about "did I miss something?".
Detection, as I said above, is merely the "Level 1 scan". If you want more money per detected planet, you need to run higher level scans. A "Level 2" scan is done by targetting the star/planet/whatever and pointing your nose right at it. A little twirly-hexagon thing appears over where it describes the object as "Unexplored". Wait for the twirly to finish twirling (takes about 15-30 seconds, depending on how far away from the object you are) and viola, you have now completed a Level 2 scan. You can now see numerous details about the planet on the System Map that you didn't know beforehand. You can target stars as well as planets for this scan, so it always pays to select and point your nose right at your arrival star (after coming to a stop, of course!) to get the extra credit; you can even be honking the horn or selecting and loading the System Map while the star scan is running.
Level 3 scans are just like Level 2 scans, only you need to have a Detailed Surface Scanner installed in a utility slot first. You don't need to arm it or assign a weapon slot for it; it just sits there and earns you more money while you scan stuff. It quickly pays for itself, as Level 3 scans are much more valuable than 2s or 1s.
I don't go anywhere without an Advanced Discovery Scanner and a Detailed Surface Scanner in my ship. That's been my rule ever since I could afford them, in my Cobra; I upgraded my scanners before I upgraded my drive.
Once you've scanned the system as much as you wish to, you can now sell the data for that system. You will have to travel more than 20 LY away from there in order to sell it, but further distance does not generate extra money. This data is saved on your ship's computer up until you sell it; that means you can still see the system you've explored on the Galaxy map after you leave it. But remember, just like uncollected bounties, un-redeemed system scans are lost forever if your ship is destroyed before you cash them in. So if you're thinking of doing something risky like accepting an assassination mission, cash in your exploration data before you go and get yourself killed.
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