Limitations of Kolmogorov–Smirnov test In this article, it is said that the test statistic tends to be more sensitive near the center of the distribution than at the tails. Can some please explain this what it means? (with a simple example if possible). Also please let me know when should the KS test test is preferred and not preferred for testing normality. 
 A: The variance of the sample cdf in the tails is smaller than near the median.
I assume we're talking about a one-sample Kolmogorov-Smirnov (though similar comments come into the two-sample case).
Specifically, the variance is proportional to $F(1-F)$, so as $F$ approaches $0$ or $1$ the variance goes to zero, and so does the standard error of the ecdf.
This means that even small deviations in the tail can be a strong indication of a problem with the null distribution. 
However, the Kolmogorov-Smirnov "sees" a given difference in the tail and in the middle as equally serious (they both give the same statistic), but it's much "harder" for the data to deviate that much in the tail so the test is relatively insensitive to what's going on in the tails compared to some other tests.
In particular, try looking at the uniform case (since any other case can be reduced to the uniform by transforming the data by the hypothesized cdf under the null). Here's 500 uniform ecdfs with n=20

If you look at the width of the spread near $x=0.5$, it's much wider than it is at $x=0.05$ or $x=0.95$. So it's "harder" to achiever the critical value of D in the tail region than in the middle -- so the test generally finds deviations more toward the middle than right up at the ends. 
