The terminology of right and left skewness is based on the idea that you are working with a display in which magnitude is plotted on a horizontal axis and also increasing from left to right, as is conventional with histograms, frequency polygons and plots of smoothed density estimates. Conventional here just means "more common" and there can be good reasons to plot otherwise. (I have published histograms of land surface altitude with a vertical magnitude scale. Nothing is more vertical than altitude.)
The term for skewness (left- or right- skewed) that is used is that of the longer tail, so broadly speaking a left-skewed distribution has a longer left tail, stretched out to low values, and a mode towards the right at high values, and a right-skewed distribution has a longer right tail, and a mode towards the left at low values.
The terminology could have been the other way round, to refer to where the main hump of the distribution lies, but it is not. Some people find the terminology backwards on first acquaintance, and that seems reasonable to me. There are occasional (even widely-selling) textbooks which flip left and right. In the most astonishing case known to me the explanation is surely utter incompetence, given the many other errors and confusions in the book, not a personal view that the terminology is the wrong way round. At a minimum, any author who regarded the conventional terms as misguided would still be obliged to explain that conventional terminology.
A terminology that fits more generally is that positive skewness corresponds to right skewness and negative skewness to left skewness, where positive and negative are the signs recorded for measures of skewness, such as
the moment-based measure defined by T.N. Thiele and Karl Pearson
the useful and neglected (mean $-$ median) / SD, which turns out to fall in $[-$1, 1$]$, or
any measure based on paired quantiles,
[(upper quantile $-$ median) $-$ (median $-$ lower quantile)] / (upper quantile $-$ lower quantile)
In #3, most commonly the quantiles chosen are the quartiles. This latter family has been attributed to Galton, who didn't use it, and to Yule and Kendall, who did use it but certainly didn't invent it. It appears to have been introduced byBowley in 1902. To see how that family of measures behaves, for concreteness consider the median and quartiles. In a highly right-skewed distribution the median can approach the lower quartile and even equal it if ties are present, so the measure reduces to (upper quartile $-$ lower quartile) / (upper quartile $-$ lower quartile), or 1. Other way round, in a highly left-skewed distribution, the median could be equal to the upper quartile, so the measure reduces to $-$1. Naturally, if upper quartile and lower quartile are equal, the measure is indeterminate, but there you go.
Whenever as often a box plot (or any other display) is plotted with a vertical magnitude axis, then right skewness corresponds to the upper tail being longer and left skewness to the opposite, although positive and negative skewness are the terms I would recommend here.
Box plots are quite often plotted with horizontal magnitude axis, in which case left and right skewed do have their traditional meanin.g
Skewness may judged informally from box plots by where the quartiles lie relative to the median and by where the extremes lie relative to the median. Most simply, a right-skewed distribution or positively skewed distribution has the upper quartile and the maximum further from the median than are the lower quartile and the minimum; and conversely for a left-skewed or negatively skewed distribution; but do read on.
There is much small print here:
It is entirely possible for a distribution to be complicated and (say) skewed one way in its middle and the other way in its far tails.
A graph and numerical measures may appear to contradict each other. A distribution can appear to be left-skewed or negatively skewed graphically and right- or positively skewed numerically, and vice versa.
Numerical measures don't necessarily agree even as to sign and in broad terms may only agree exceptionally, notably that all measures of skewness yield zero for an exactly symmetric distribution.
In all of the above, it's important to remember that (the appearance of) skewness is influenced by individual outliers as well as the broad shape of the distribution and it can difficult to separate the two. However, one merit of a box plot is certainly to show up cases in which what is going on between the quartiles has a different flavour from what is going on between the extremes. The same is, however, also true of most displays of distributions that preserve most or all of the detail of a distribution.
Box plots can give very odd-looking patterns if there are many ties and/or if there is bimodality or multimodality. Box plots from Likert scores or counted data with smallish integer values can give rise to many puzzled reactions, for example.