The quick version of the new gymnastic scoring system is this:
There is both a start value (based on difficulty), and an execution score (based on making mistakes).
The start value essentially starts at zero. As the routine progresses, more difficult feats garner more points, either because of their inherent difficulty, or because they're done in combination with other feats. Doing a back flip, stopping, and then doing another back flip is easier than doing a back flip and immediately flipping again once you've finished, so there's a bonus for doing the two in connection with each other. Only the top x moves count in determining start value - I think x is ten in most events, but it's possible that it changes. The start values they show you before a routine are what the routine should get as a start value if the gymnast does everything as expected, but this can be altered if the routine is changed at all--doing a single twist instead of two, or not doing a combination because you faltered on an earlier part of it. There are several judges who simply add up whatever the athlete does and record the start value.
The execution score starts at a 10.0, and goes down for every mistake. Steps on the landing, stepping out of bounds, falling from the equipment, separating the legs when they're supposed to remain together, not holding a handstand for long enough, etc.
In the old days, there were start values too, they were just much less well publicized. Old start values were either an 8.4 or 8.8 (depending on if it was men's or women's) + something for the difficulty of the moves performed. The problem was that the gymnasts had reached a level where essentially all the major competitors were able to do routines with start values of 10 or greater. This meant that there was no longer any incentive to try more difficult things than were already being done, as a start value above a 10.2 didn't help--and you only really wanted the extra .2 as insurance in case you weren't able to do something as expected. From that start value (or 10.0, whichever was lower), deductions were taken until you reached a final score.
Conceptually, I like this new scoring system better than the old one (though not as much as I like diving's), but it needs to be tweaked. One of the major things it tried to do was equalize the values across appartus, and that has obviously failed. Also, some of the deductions, in my mind, are simply not strong enough, leading to a bronze medal on vault going to a girl who landed on her knees even though there were others who did difficult vaults and landed on their feet and who didn't medal. yes, it's important to reward individuals for trying more difficult things, but it's also important to reward individuals for doing things correctly, and I don't think they've struck the right balance yet.