If I look at a quantile plot comparing normal distribution and other data, and some of the points lie below or above the line, how do I know what that represents? What I am asking is - how do I know if the data set has a left or right skew?
Well, you wouldn't know because both tails are effected by skewness, kurtosis and other moments. For example, and although an accepted answer leads to an examination that only modifies skewness and kurtosis (called tailedness by the author). This does not answer the question as it leaves out all higher moments AND the role of outliers upon Q-Q plots. Indeed, the influence of higher moments and outliers is not even discussed. To say this another way, a normal Q-Q plot can be used to examine data that is not normally distributed, and in so doing, the skewness is only indirectly shown. One method of determining skewness is to just calculate it.
Positive skewness has been said to have a longer or fatter right (than left) tail. Fat and long tails are not quite the same things, such that the usual graphical explanation of skewness is somewhat ambiguous. For example, we could have a fat left tail and a long right tail, and that is not uncommon. Thus, the best definition of skewness is from the formula used to calculate it. That is, although we plot what skewness looks like on a histogram or Q-Q plot in particular cases, there is no unique graphic description of skewness, only a mathematically one. This lack of uniqueness prevents us from inverting the problem and determining skewness from a graph, except in cases that have been so narrowly defined, that that graphic interpretation is unique.