Stokes flow

Since $216=3^{3}+4^{3}+5^{3}=6^{3}$ , it is the smallest cube that's also the sum of three cubes (Plato was among the first to notice this, and mentioned it in Book VIII of Republic). It is also the sum of a twin prime (107 + 109). But since there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number.

This multiplicative magic square

has magic constant 216.

In base 10, it is a Harshad number.

There are 216 fixed hexominoes, the polyominoes made from 6 squares.

In other fields

The 216 colors formed by a 6×6×6 color cube are sometimes known as the web-safe color palette.
ISO 216 is the ISO's standard for paper sizes.
In the film Pi, Max becomes obsessed with a 216-digit number in his quest to find the numeric equation behind existence, and is told by a group of Hasidic Jews that it represents the true name of God.
216 is the numerical value of Gevurah, the fifth sephirot in the kabbalistic tree of life, and the sum of the letters in God's 72 hidden names.
As six cubed (6*6*6), 216 is also symbolic of the Mark of the Beast in Christianity. This has been expressed in the Left Behind series.
According to the Ken Burns series Baseball, it is the number of stitches on a baseball.
+216 is the country code of Tunisia.
(216) is the area code of Cleveland, Ohio.
The Bayon temple in Cambodia has 216 gigantic smiling faces.