Koyasan is a very sacred place in Japan, as it was where the monk named Kukai (Kobo Daishi 弘法大師) founded Shingon Buddhism. Kukai was the first to settle the mountain in 819, and from there it grew into the town of Koya as it is known today. Koya is home to many temples and shrines, including Okunoin (奥の院), which houses Kukai's mausoleum and is surrounded by the largest graveyard in all of Japan.
The town offers temple lodging (shukubo), where visitors and pilgrims can opt to stay overnight and experience local vegetarian cuisine and meditative practice. Shown in the photos below is Fukuchiin (福智院), a shukubo with beautiful interiors and impeccable landscaping. I recommend staying at least two nights, if you can.
Koyasan is popular for its brilliant autumn colors as well. With the brisk mountain air and transient fog, the place becomes alive as you walk through the somber, quiet forests and temples. It's a different world there.