I have histograms/distributions like the following that I would like to characterize/summarize by one or maybe two number(s). A mean +/- std would be one such example, but is there something better that can say something about the shape of the distribution? Most of the times the most frequent values are near the two ends, so the distributions are not normal. Is there a statistic for being the opposite of normal distribution?
Could it be that your data follows the family of U-shaped distributions? https://en.wikipedia.org/wiki/U-quadratic_distribution
The simple answer is: You can't.
Slightly more complex answer: The reason you can summarize a Normal distribution with two numbers is that, once you have the mean and the SD of a Normal distribution, a lot of the rest is determined. While there can be variations in any particular sample, the skewness of a Normal is 0, the excess kurtosis is 0, it's symmetric and you can estimate the number that will be in any bin of a histogram.
If you are willing to assume another distribution, then you may be able to summarize the data with one, two, three or four values, depending on the particular distribution. For instance, a Poisson distribution can be summarized with one value.
However, if you are not willing to make that assumption, then any summary will lose a lot of information. You could use range or interquartile range for spread, and median or another trimmed mean for location, but this will be much less informative than mean and sd for a Normal.