If you imagine a histogram of a distribution that resembles a "normal" (or 'bell') curve as being like a diagram of a mountain, a curve/mountain that is unevem and squashed up against the Y axis is said to be positively (or right) skewed: that is, there are more cases in the lower 'bins' of the histograms than the higher ones. Note that the 'direction' of the skew (left or right) refers to the tail or slope of the mountain, not the peak!
A curve that looks as though it has been pushed in the opposite direction, i.e. away from the Y axis, is said to be negatively (or left) skewed: there are more cases in the higher bins.
A mountain that is very tall and thin is called 'leptokurtic' or having high kurtosis, with a lot of cases in the middle bins: a mountain that is flat and wide is 'platykurtic' or with low kurtosis, with cases spread quite evenly across the bins. One that is in the middle (like the normal distribution curve) is 'mesokurtic' or having average kurtosis.
The explanation isn't super brief, but hopefully it's more clear than some of what you've encountered in the literature.:
Skewness means that the distribution of data you have isn't symmetrical (see a normal bell curve—the right and left portions look like mirror images of one another; a skewed distribution is lopsided). A positive skew means that a whole bunch of cases are piled up on the left side (think a lump of scores occurring on the left side rather than being in the middle, like a normal curve) and the tail, or skinnier part of the distribution is longer and extends in the positive direction (so the tail extending in the positive direction is why it’s called a positive skew). A negative skew is the opposite—big lump of scores occurring on the right side and the tail going to the left—in the negative direction (like the distribution of scores on a test that way way too easy). Skew can be a problem because extreme skewness can make parametric statistics, and the conclusions we make based on them, less precise.
With kurtosis, the curve may be symmetrical (i.e., not necessarily skewed) but the distribution has either more or less dispersion of scores compared to a normal distribution. Platykurtic means that the scores are spread out more than normal and that there are more extreme cases at the two tails of the distribution (it looks like a spread out and flattened bell curve—like someone squished a taller pile of dirt into a flatter one). This makes standard deviations (an index of dispersion) relatively large because scores are so spread out. A too restricted distribution has most of the scores clustered in the middle and way fewer scores at the tails than normal (looks more like a tower). This is called a leptokurtic distribution. SDs in this instance are often quite small relative to if the distribution of scores (or other data) were more normal. In the extreme, this can also affect statistics and conclusions in an undesirable way. It’s important for researchers to investigate distributional problems and deal with them in a way that lets the reader understand what the problem is/was, how the researcher choose to deal with it, and any implications these distributional problems have relative to the conclusions the researchers are trying to draw, based on the data. So skewness is a problem with a distribution’s symmetry, and kurtosis is a problem with its dispersion. Look for pictures online by going to Google images—that will also help make these concepts more clear. Some distributions have problems with both skewness and kurtosis (not to mentioned outliers—very extreme scores--which are the third issue to consider when reviewing data distributions). Hope this helps.