Fully described by mean and variance. About 68% of mass within one standard deviation, 95% within two, 99.7% within three.
It appears everywhere because of the central limit theorem: sums of many independent finite-variance contributions tend to it regardless of their own shape.
Sums of independent normals are normal, which makes it uniquely tractable.
The trap in finance. Returns are not normal - they have fat tails and negative skew. Using normality for tail risk understates crash probability by orders of magnitude, which is where the "we saw multiple 20-sigma moves" complaints come from.