In a normal distribution, the 68-95-99.7 rule imparts standard deviation a lot of meaning, but what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions?
It's the square root of the second central moment, the variance. The moments are related to characteristic functions(CF), which are called characteristic for a reason that they define the probability distribution. So, if you know all moments, you know CF, hence you know the entire probability distribution.
Normal distribution's characteristic function is defined by just two moments: mean and the variance (or standard deviation). Therefore, for normal distribution the standard deviation is especially important, it's 50% of its definition in a way.
For other distributions the standard deviation is in some ways less important because they have other moments. However, for many distributions used in practice the first few moments are the largest, so they are the most important ones to know.
Now, intuitively, the mean tell you where the center of your distribution is, while the standard deviation tell you how close to this center your data is.
Since the standard deviation is in the units of the variable it's also used to scale other moments to obtain measures such as kurtosis. Kurtosis is a dimensionless metric which tells you how fat are the tails of your distribution compared to normal
The standard deviation is one particular measure of the variation. There are several others, Mean Absolute Deviation is fairly popular. The standard deviation is by no means special. What makes it appear special is that the Gaussian distribution is special.
As Pointed out in comments Chebyshev's inequality is useful for getting a feeling. However there are a more.
The sample standard deviation is a measure of the deviance of the observed values from the mean, in the same units used to measure the data. Normal distribution, or not.
Specifically it is the square root of the mean squared deviance from the mean.
So the standard deviation tells you how spread out the data are from the mean, regardless of distribution.