Is there any statistic or method which could describe how much a distribution deviates from the normal distribution? I want to test if my data is normal or not at some level of significance, and I have huge many of samples, I am wondering if there is any quantitative way to test it on my computer. Thank you very much.
Skewness and kurtosis are helpful statistics to see how far from non-normal you are. A normal distribution should have a skew near zero and a kurtosis near three (note that some programs subtract three from kurtosis, so kurtosis should be near zero, as well. You'll have to check documentation).
Here are the statistics for a randomly-generated normal distribution:
> set.seed(1839) > data <- rnorm(1000) > moments::skewness(data)  0.03057696 > moments::kurtosis(data)  2.95321
Here's what it looks like for a poisson distribution:
> set.seed(1839) > data <- rpois(1000, 0.1) > moments::skewness(data)  3.129368 > moments::kurtosis(data)  12.08103
You can also look at a histogram, where you want it to look like the typical bell-curve, or a Q-Q plot, where you expect the data points to fall on a line where the observed quantiles are equal to the expected quantiles.
There are a number of ways to test for "significance" of non-normality, but I would warn against doing this, especially if you have large samples. Even a small deviation from normality can be significant when $n$ is large enough (see simulations here).
Given that many methods can handle some level of modest non-normality, I would use your eyes: Look at the Q-Q plot, look at the skewness and kurtosis. I would not rely too much on a $p$-value if you have a large $n$.
If you are worried about normality, you can always use other robust methods such as sandwich estimators or bootstrapping.
You can use any Normality test. Kolmogorov-Smirnov, Liliefors and Shapiro–Wilk test are usually implemented in statistical softwares. I would use the Shapiro–Wilk test. The value of the test statistic can be used as the maesure of non-normality.