Mount Hiei (比叡山 Hiei-zan),  is a mountain that lies to the northeast of Kyoto in the Sakamoto (坂本) region of Otsu. It is famed for the extensive temple complex near the summit. Since many travelers come here from Kyoto, they often enter from the town of Yase (八瀬) at the base of the mountain on the Kyoto side.
The temple of Enryakuji, the first Japanese outpost of the esoteric Tendai sect of Buddhism, was founded atop Mt. Hiei by Saichō (Dengyō Daishi) in 788. The temple complex was razed by Oda Nobunaga in 1571 to quell the rising power of the Tendai's warrior monks, but it was rebuilt and remains the Tendai headquarters to this day.
There are several ways to reach Sakamoto and Mount Hiei.
From Kyoto, the JR Kosei line goes directly to the Hieizan-Sakamoto station which is a 1,200 meter walk from the Sakamoto Cable Car or the trailhead to climb Mount Hiei on foot. Alternatively, take the Keihan Main Line to Demachiyanagi and transfer to an Eizan train to Yase-Hieizan-guchi (八瀬比叡山口). From here the Eizan Cable Car makes the trip to the top of Mount Hiei for ¥530/1040 one-way/return (incl. the ropeway: ¥820/1640 one-way/return), every 30 minutes daily from 8:30 to 17:30 (or longer, schedules vary a bit depending on the season). The last leg of the trip to the summit is a 3-minute ride on a ropeway, which departs at intervals of 10-20 minutes between 9 am and 6 pm.
Alternatively, you could also take the Keihan Line to Sanjo Station and transfer to the Tozai Line bound for Hamaotsu Station in Otsu. From Otsu, you can take the JR Kosei line or Keihan Ishiyama-Sakamoto line to Sakamoto, although the Keihan station (the last station on the line) is more centrally located. The Hiyoshi Taisha shrine and the cable car to Mt. Hiei are about 15 min away on foot, both fairly well signposted.
Be careful: there are two different cable-cars up the hill, each ran by a different company. So, if you buy a two-way-ticket from one company, you cannot use this for the other railway and you'll need to buy a new ticket.
There are occasional direct buses from Kyoto station directly to the top, taking about 1.5 hours and all departing in the morning. Schedules are severely curtailed in the winter.
Both Sakamoto and Mt. Hiei are best covered on foot. For going between the two, you can use the Sakamoto Cable Car, which costs ¥840/1570 one-way/round-trip and runs daily from 8 AM to 5 PM once every 30 minutes. At over 2 km, this is the longest cable car in Japan and takes about 11 minutes for the journey. This cable car line was built in 1927 and refurbished in 1993. The European style cars have large windows with wonderful views of Lake Biwa.
A real pilgrim would of course scoff at mechanical contraptions and climb the mountain, which is fairly easy as this isn't really more than an oversized hill. The traditional route is a convenient path of mossy steps known as Honzaka (本坂), starting from Sakamoto, but it's still 500 meters (vertical) to the top. There are also many other routes, with numerous small temples and waterfalls along the way, but watch out as signposting (even in Japanese) is lacking. You may see monkeys along the way as well.
Both the Sakamoto cable car from Lake Biwa side and the Hieizan Ropeway from the Kyoto side terminate near two broadcast towers and the Garden Museum Hiei, whose ticket booth has maps to Hiei's more traditional attractions, which can be reached by about twenty minutes' walk through atmospheric forest.
The Garden Museum features spectacular views of Lake Biwa, garden flowers, art galleries, a cafe, large outdoor reproductions of famous works of French Impressionism, and a lily pond with arched bridge that aims to replicate the Japanese garden designed by Claude Monet in Giverny, France and featured in many of his late paintings. It also houses the grave of Saichō (Dengyō Daishi) the founder of Japan's Tendai Buddhism, and attracts many Japanese bush warblers (uguisu). The entrance fee is ¥1000 for adults, ¥500 for children.
The temples on Mt. Hiei are collectively known as Enryakuji (延暦寺), literally "Long Calendar Temple". The large complex is generally divided into three sections known as the Eastern Pagoda (東塔 Tōdo), the Western Pagoda (西塔 Saito) and Yokawa; neither of the pagodas actually exist any more, but the names live on. Most of the better-known temples are concentrated in the Eastern Pagoda area.
There is a wide range of accommodation in Sakamoto, but many visitors choose to day-trip from Kyoto instead.