Interpretation of Standard Deviation independent of the distribution?

The standard deviation measures dispersion of a random variable regardless of the distribution. Chebyshev's theorem states that any random variable (whether normally distributed or not) takes values more than $k$ standard deviations from the mean with probability less than $\frac1{k^2}$.

For example, it is true for any distribution that at most $\frac19$ of the values are more than 3 standard deviations from the mean. Or turning it around, for any random variable, at least half the values lie in the range $(\mu - \sqrt2\sigma, \mu +\sqrt2 \sigma)$.

In most cases, the Chebyshev bound is extremely loose, and more of the distribution is closer to the mean than it would suggest.