IF the "curve" is "normal", with some mean and standard deviation, there is a relationship between the percentages at different lengths along the absissa.
Now, if you think about it, there are "many" normal curves with various means and standard deviations. It wouldn't be possible to relate those percentages to the various means/s.d. unless you could "standardize" those numbers. We thus "standardize" those numbers and call them "z scores", which we can then table. We standardize them by the following formula:
z = (score - mean) / s.d.
Let's look at a distribution with mean = 100, s.d. = 20. A score of 120 would have a "z score" of:
z = (120 - 100) / 20 = 20 / 20 = 1.00
If another distribution had a mean of 50 and s.d. of 10, a score of 60 would have a z score of:
(60 - 50) /10 = = 10/10 = 1
In some sense, a score of "120" on one distribution is "equivalent" to a score of 60 on another.
Now, there is one number you must memorize. 1 z score is the 84th percentile, which is 34% above the mean which is 50%. This memorization will allow you to use any z table (normal curve table). You see, different tables report the percentages different ways. When using any table, go to the z score of "1" and look at the number. If it is "84", the table reports the percentages from the left tail; if it is "34", the table reports the percentage from the mean.
Now, we know the curve is symmetrical; and we know the mean is the 50th percentile; so let's play with it. If 1 s.d. is 34%, then, it must be the 84th %ile, because we have to add the 50% in for the mean. If 1 s.d. is 84, then there must be 16% ABOVE 1 s.d. Going the "other way", -1 s.d. must be 34% from the mean, or 84% from -1 s.d to the "right edge", so it must be the 16th %ile.
You should "play" with some more values, flipping back and forth above the value, below the value, using the left side of the mean and the right side of the mean. Drawing pics of the situation helps.