How to determine whether data is slightly or extremely non-normally distributed? I'm a PhD student and doing a research on regression analysis.
My question is how to determine whether the data is slightly, moderately or extremely non-normally distributed?

TQ to all responses of my question. But, may be my question is not so clear.
Ok, let say i have some different values of skewness and kurtosis (for example: skewness = 1.5, kurtosis = 2.0). So my question is, from the values of skewness and kurtosis, what is the type of data distribution? is it moderately non-normal, or extremely non-normal or what?
 A: The sample skewness $$\gamma=\frac{\sum_{i=1}^n(x_i-\bar{x})^3}{\Big(\sum_{i=1}^n(x_i-\bar{x})^2\Big)^{3/2}}$$ and the sample (excess) kurtosis $$\kappa=\frac{\sum_{i=1}^n(x_i-\bar{x})^4}{\Big(\sum_{i=1}^n(x_i-\bar{x})^2\Big)^{2}}-3$$ are often used as measures of non-normality.
The sample skewness measures the asymmetry of the empirical distribution. If it is far from $0$, the distribution is not very symmetric. Since the normal distribution is symmetric, a sample from the normal distribution should be close to $0$.
The sample kurtosis measures the "peakedness" of the distribution. If it is much greater than $0$, then the distribution is more peaked than the normal distribution, which typically means that it has heavier tails. If it is less than $0$ it is less peaked, which typically means that the distribution is bimodal. The sample kurtosis is bounded from below by $-2$ (a value that is obtained for a two-point distribution, which of course is extremely bimodal).!
Here are two examples (normal distribution in grey, other distributions in red):

The skew distribution has theoretical skewness $1.6$ whereas the kurtotic distributions has theoretical (excess) kurtosis $1.5$. As you can see, the kurtotic distribution has heavier tails than the normal distribution.
So, why use skewness and kurtosis as quantifications of non-normality? The main reason is that they affect the asymptotics of the central limit theorem, which as you may know often can be used to motivate the use of a statistical procedure (that is based on normality) even if the data does not come from a normal distribution, given that you have a "large enough" sample. If either the skewness or the kurtosis is high, larger sample sizes are needed for such motivations to be valid.
For some inferential procedures you need to worry more about skewness, and for some you need to worry about heavy tails (kurtosis). I've written more about that elsewhere on this site.
A: I think the other answers really address methods for deciding non-normality.  But I think the OP asks a different question.  Basically he is asking about how to decide once normality is rejected how do you determine the severity?  If it is mild perhaps the deviation can be ignored.  Skewness and kurtosis can be looked at as measures of non-normality but I think it comes down to a subjective decision as to how large a difference should be to call it, mild, moderate or large.  I think the point is that this decision can be made by looking at histograms, qq plots, or the magnitude of the skewness and kutosis.  But this is going to be subjective and not formal.
A: Although a wikipedia link might be not considered extremely helpful, the list of methods for testing normality is quite long.
The methods range from (already mentioned) histograms and qq-Plots if you want to stay on the graphical side over more lets say "empirical tests" (multiple sigma events in relation to sample size) to parametric and nonparametric statistical tests. A complete review of these would be, in my opinion, out of scope here (and for some methods definitely also out of my scope) so I will be quite frank here.
Since you were mentioning regression analysis I guess you want to test the normality of the residuals. Just use one of the normality tests on the wiki page. The more popular variants compare skewness and kurtosis to that of a normal distribution. The nonparametric versions are Kolmogorov-Smirnov type of tests that use the empirical cumulative distribution function of your data (probably the residuals). Just look at the wiki page. The standard tests are quite simple to implement.
A: The Skewness of the Normal distribution is 0. The Kurtosis of the Normal distribution is 0. Those two statistics are concrete measures of distribution characteristics as opposed to subjective plot interpretation. Of course the question remaining is how far from Normal your distribution is for $n$ points and some value of Kurtosis and Skewness.
Hence you will be better off by running some Normality Tests. The Shapiro-Wilk is a sensible choice for univariate data.
If you use R you will find the function shapiro.test() in the stats package useful. The moments package includes the functions skewness() and kurtosis() plus jarque.test() if you want a second opinion after the results of the Shapiro-Wilk test.
A: Here I found some rules of thumb on how to evaluate. 
If skewness is:


*

*less than −1 or greater than +1, the distribution is highly skewed.  

*between −1 and −½ or between +½ and +1, the distribution is moderately skewed.  

*between −½ and +½, the distribution is approximately symmetric.  

