I do know there are some rules about boxes and whiskers to determine the skewness in a boxplot, but I am confused with some rules in this particular case:
Keeping in mind the rules, in this boxplot the median falls to the right of the center of the box, thus its distribution is negatively skewed. But I also can see that the right line is larger than the left line, thus "according to the rules" the distribution is positively skewed. How do I know the real skewness. Thanks in advance.
If you have access to the underlying data you could/should estimate the skewness according to some measure, a common measure being based on the third moment of the data, see here. Note though there are other measures and they won't necessarily give the same results.
Regarding the box plot itself, it is a crude summary of a distribution and not necessarily good enough to allow the skewness to be estimated. Indeed, there can be quite different data sets, with quite different skewness, that will have the same boxplot.
You are correct that 'indications' of right-skewness of a sample from a boxplot may be that (a) the median is left of center inside the box and (b) a longer whisker to the right than to the left. However, boxplots are best used for samples of moderate or large size.
Of course, I don't know for sure, but I would guess that the contradictory indications in the boxplot you show are likely because the sample size is small. (You might use some mathematical measure of skewness as the Comment by @scitamehtam (+1) suggests, but as mentioned there various measures of skewness can give different results, and this is also especially likely to happen with small samples.)
Below are boxplots of 20 samples of size $n = 15$ from a normal population. The normal distribution is symmetrical, so you might suppose the boxplots would not show skewness. But there are all sorts of indications of skewness: medians not in the centers of boxes, and whiskers of noticeably different lengths.
By contrast, here are boxplots of 20 samples of size $n = 1000$ from the same normal population. These boxplots do not show such conflicting results about skewness; most of them are consistent with data from a symmetrical distribution. (Don't worry about the 'outliers': They are to be expected in boxplots of large normal samples because the 'tails' of the normal distribution extend to $\pm \infty.$)
Finally, here are boxplots of 20 samples of size $n = 1000$ from a (severely right-skewed) exponential distribution. All of them have the indications of skewness you mention. (They also have lots of outliers on the high side, and none of the low side; another indication of data from a skewed population.)
