σ — standard deviation, singular value, or sigmoid

$\sigma = \sqrt{\operatorname{Var}}$. Taking the square root puts the spread back in the same units as the data, so you can compare it with the values themselves. $\sigma^2$ is the variance.

In Section 20.2 the same letter is used for singular values $\sigma_1 \ge \sigma_2 \ge \dots$, which say how strongly a matrix stretches along each of its main directions. The two meanings are unrelated; a numbered subscript ($\sigma_1$, $\sigma_i$) usually means a singular value.

Written as a function, $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid, which squashes any number into the range 0 to 1. That is its meaning in Swish, $z \cdot \sigma(\beta z)$, in section 15.4: the sigmoid acts as a soft on/off switch on $z$.