The first perfect ten in Olympic history was scored by Nadia Comaneci of Romania, on uneven bars during the compulsory round at the 1976 Olympic Games in Montreal. The last perfect ten in Olympic history was scored by Lavinia Milosovici, also of Romania, in the floor exercise event finals at the 1992 Olympic Games in Barcelona. Now, gone is the perfect ten. Starting in 2006, the FIG unveiled a brand new scoring system. This new scoring system continues to confuse people, especially casual fans who only watch gymnastics during the Olympics. I'll do my best to break it down in the plainest English possible so everyone understands.
The D-Score
The new system consists of two numbers, a D-score and an E-score. The D is for difficulty, which is the start value of the routine. It's calculated from the value of each skill performed plus five tenths for every requirement on each apparatus, and connection bonus. Each skill in the Code of Points has a value from A to G (or H on floor exercise), which is translated into tenths.
A = 0.1
B = 0.2
C = 0.3
D = 0.4
E = 0.5
F = 0.6
G = 0.7
H = 0.8
The top 8 most difficult elements are added together, with the connection bonus and 2.5 points for fulfilling each requirement to create a start value.
BARS REQUIREMENTS:
flight from high bar to low bar = 0.5
flight from low bar to high bar = 0.5
flight element on same bar = 0.5
close bar elements = 0.5
dismount D+ = 0.5
For beam and floor, of the top 8 elements counted towards the start value, five must be acrobatic skills, and three must be dance skills (leaps, jumps, and turns).
BEAM REQUIREMENTS:
connection of two dance elements (one must be a 180 degree split) = 0.5
turn on one foot = 0.5
acrobatic series = 0.5
acro elements forward & backward = 0.5
dismount D+ = 0.5
FLOOR REQUIREMENTS:
connection of two dance elements (one must be a 180 degree split) = 0.5
acrobatic series = 0.5
saltos forward & backward = 0.5
twisting & double back saltos = 0.5
dismount D+ = 0.5
Every vault already has a start value that does not need to be calculated.
Let's take Aly Raisman's floor exercise from the Olympics, for example.
First pass: 1½ twist (C) + Arabian double front (E) + front layout (B) (connection bonus: 0.4)
Second pass: Arabian double pike (F) + sissone (A) (connection bonus: 0.1)
Two connected dance elements: switch split ring leap (C) + split leap 1½ (D)
Dance element: full turn (A)
Third pass: triple twist (E)
Dance element: switch split leap with full turn (D)
Fourth pass: double pike (D) + split jump (A) (connection bonus: 0.1)
The top 5 acrobatic elements would be:
Arabian double pike - F
triple twist - E
Arabian double front - E
double pike - D
1½ twist - C
The top 3 dance elements would be:
split leap 1½ - D
switch split leap with full turn - D
switch split ring leap - C
Acro skills: F + E + E + D + C
Dance skills: D + D + C
0.6 + 0.5 + 0.5 + 0.4 + 0.3 + 0.4 + 0.4 + 0.3 = 3.4
Connection bonus: 0.6
3.4 + 0.6 = 4.0
4.0 + 2.5 = 6.5
This makes her start value a 6.4. In the current Code of Points, it would be a 6.4, as there's no more 0.1 bonus for D (acro) + A (dance) connections.
Start values vary for each apparatus. For example, on vault, the start value for Amanar (2½ twisting Yurchenko) is 6.3. The hardest vault in the Code of Points is a handspring double front, which is a 7.0.
To medal on bars, the start value needs to be in the high 6's or low 7's. Beth Tweddle, Aliya Mustafina, and He Kexin had start values in the low 7's on bars in the 2009-2012 quad.
Beam and floor are pretty similar. To medal, the start value needs to be in the low to mid-6's. To my knowledge, Sarah Finnegan had the highest start value on beam in the previous quad - 6.8. Katelyn Ohashi has a 7.3 start value on beam now. Aly had the highest start value on floor in the previous quad.
The E-Score
The E is for execution. The execution is out of a ten, which is simple enough to understand, so I don't need to explain that. These days, an execution score in the 9's is not very common in World Championship or Olympic gymnastics. Mostly, execution scores will be in the high 8's, particularly on bars and beam.
With the new scoring system, no one has been awarded a perfect score. A few have come close. Nastia Liukin and He Kexin were the only gymnasts to score in the 17's on uneven bars during the 2005-2008 quad. McKayla Maroney’s near-perfect vault in team finals in London had a 9.733 in execution.
And Finally
Let's go back to Aly Raisman's floor exercise. Her start value (in the previous Code of Points) was a 6.5. Her execution score during event finals at the Olympics was 9.100. Add them together and you get 15.600. That is her final score.
Solid scores start at about the mid-14 range. Scores in the 15's are very good and scores in the 16's are excellent. Anything lower than 14 is considered a miss.
Scores tend to vary with each apparatus. Back in the 2001-2004 quad, vault was the lowest scoring event. Now it's floor exercise, due to the deductions gymnasts take for stepping back on their tumbling passes and pausing in the corner too long before a tumbling pass.
For the TL;DR Crowd
If you completely skipped over everything (I don't blame you, it's confusing):
Scores in the mid to high 14's are solid and consistent. The middle of the pack.
Scores in the 15's are very good. 15's win medals.
Scores in the 16's are exceptional. 16's win gold medals.
Anything lower than 14 is not good.
If a gymnast falls, they can still medal if their start values are high enough (see: Vanessa Ferrari 2006 AA, Li Shanshan 2007 BB EF, Cheng Fei 2008 VT EF, Rebecca Bross 2009 & 2010 AA, Yao Jinnan 2011 AA, Aliya Mustafina 2012 AA, McKayla Maroney 2012 VT EF).