The first full moon will happen on Night 11.

After the first full moon, one will occur every 20 days.

First Full Moon: Night 11
Second Full Moon: Night 31
Third Full Moon: Night 51
Fourth Full Moon: Night 71
Fifth Full Moon: Night 91
Etc...

Full moons are very important events. Here are some of the most important things to remember about them.

Most Important things:

*During full moons, everything will be clearly illuminated (at night) and you will not need to worry about the darkness attacking you.
*Any pigs caught outside during a full moon will transform into aggressive were-pigs.
*Full moons are your only opportunities to collect glommer's flower (which is the item to control the location of glommer, who will spawn after the flower appears) from glommer's statue. You can always find glommer's statue near the pig king.
*Aggressive ghosts will emerge from any type of grave during a full moon.
*If you repair a marble sculpture with a suspicious marble, you can mine it during a full moon to receive a sketch, marble, and a corresponding clockwork that wants to kill you. Note that mining these during new moons will get you a shadow piece instead of a clockwork.

Other Things:

*Woodie will transform into a werebeaver (or a weremoose or weregoose) during a full moon.
*Chester can transform into Shadow Chester or Snow Chester if the right conditions are met during a full moon. (Fill every slot with nightmare fuel or ice gems respectively.)
*Wild and aggressive pigs (wild boars?) will also turn into were-pigs when caught outside during a full moon.
*Tides (in beach biomes) are much stronger during full moons.