If the mean is 50 and the standard deviation is 10.
I am told that this means that "most of my data" falls within 10 points of 50. So "most of my data" is between 40 and 60.
But what does this "most of my data" mean exactly? Does it mean 51% of my data? 60% of my data?
Assuming a normal distribution it means ~ 68% of the data fall within 1 standard deviation
Here's a picture picture http://www.regentsprep.org/regents/math/algtrig/ats2/normal67.gif
Standard deviation is defined as the square root of the variance.
$$\sigma_X = \sqrt{\sigma_X^2}$$
Variance is defined as follows:
$$\text{Var}(x) = \sigma_X^2 = \mathbb{E}\left[\left(X - \mu_X\right)^2\right]$$
That is, it's a measure of how much of a spread exists between the data $X$ and its mean $\mu_X$.
This is it's precise definition. Intuitively, we say that if the standard deviation is small then the data is not spread very much, and if the standard deviation is large, then the data is spread a lot.
If we further know that the distribution is Gaussian (a.k.a. normal) then we can make precise statements about how much data falls within so many standard deviations.
