Interpretation of standard deviation if data is not normally distributed This is very basic question. But I want to know how one can interpret the standard deviation if data is not normally distributed. 
My concern is regarding financial market. Investors generally use standard deviation(like standard deviation of log returns) as an proxy for risk for financial assets.In reality distribution of return is more peaked at the center and higher mass in the tail as predicted by the normal distribution. If this is true than extreme observations will occur more frequently than predicted by normal distribution. 
Is standard deviation an appropriate measure of risk in such scenario?
Edit: The above calculated standard deviation is used for many purposes like construction of portfolio, comparing the performance of various stocks etc.
 A: The standard deviation can be interpreted as a summary measure that captures variability. It is measured in the same units as the original variable. 
With additional assumptions, other interpretations are possible. For instance, if n random variables are statistically independent with standard deviation $\sigma$, then their average has standard $\sigma/\sqrt n$, and because of the central limit theorem, it gets closer and closer to normally distributed as $n$ increases.
For some random variables, these assumptions are not satisfied. In financial markets, one should expect certain random variables to be correlated: if you see 3,999,999 foreclosures, expect number 4 million soon. This may prevent standard deviation from decreasing with $\sqrt n$ as many observations are averaged. 
In addition, some heavy-tailed random variables have infinite standard deviation, and any finite-sample description or inference using a standard deviation becomes misleading. You could instead consider the expected absolute deviation $E[|X - \text{median}(X)|]$, which always exists.
A: 
Investors generally use standard deviation(like standard deviation of
  log returns) as an proxy for risk for financial assets.

This is a blanket statement that is probably incorrect. You could say "some investors" to make it more reasonable. You can look up risk measures in Google. There are many different ones used. For instance, it was noted that people are more averse to losses compared to gains. This would make a symmetrical measure of risk inappropriate in some situation. In some cases absolute deviations are used instead of squared ones etc.
The basic textbooks in MBA programs would only cover Markowitz' model, of course, leaving them with an impression that it's all that investors do. There's this whole cottage industry of portfolio optimization not based on Markowitz's mean-variance approach, but using other measures of risk. 

In reality distribution of return is more peaked at the center and
  higher mass in the tail as predicted by the normal distribution.

It's very well known that equity returns are not Gaussian, and many other financial series too. If you Google "fat tails returns" or something like that, you'll get a ton of papers on the subject. The investors, at least sophisticated ones, know this very well and for a long time. Those who didn't pay attention, surely, got the message delivered in 2007.
Also, as statisticians will tell you the standard deviation can be computed on any distribution, not only Gaussian. It's a measure of dispersion of the random variable. As long as you don't stick "Gaussian" p-values to it, it's not misinforming you at all on fat tailed distributions. For this reason, mean-variance framework for portfolio optimization is not necessarily inappropriate for fat tailed distributions. It can be inappropriate for other reasons, such as asymmetrical risk aversion or non-quadratic risk-aversion.
