Shemini Atzeret has various customs, including reciting the memorial prayers of Yizkor. And the tradition offers beautiful supplicatory prayers to recite on this day which ask God for rain. (I wrote a contemporary one several years ago, in the form of a ghazal.) But my favorite teaching about today is that today is a day for sitting still.
The Hasidic rabbi known as the Slonimer Rebbe teaches that there are two days called atzeret, "pausing," during the year. On each of these days, God asks us to be people who choose to pause, to linger in the divine presence.