Sensoji (浅草寺) is one of the most well-known Buddhist temples not only in Tokyo but in whole Japan. Asakusa Jinja (浅草神社) is a Shinto shrine next to it and used to be almost part of the temple until the late 19th century. (See the article on Sensoji Temple & Asakusa Shrine for details.)
According to legend, Sensoji was founded on March 18, 628 CE. One morning, fisherman brothers named Hinokuma Hamanari and Hinokuma Takenari found a statuette in their fishing net. They let a man called Haji no Nakatomo see it, and he realized that it was a figure of Sho-Kannon. He built a hut and enshrined the statue there, and this hut was the foundation of Sensoji. The Honzon-jigen-e (本尊示現会) is a ceremony to commemorete this event. A honzon (本尊) is the main idol of a Buddhist temple, jigen (示現) means the appearance of a deity or its supernatural power, and e (会) is a suffix which is used to form the name of a Buddhist ceremony. Until the then new government separated Shintoism and Buddhism in 1872, the Sanja Matsuri of Asakusa Shrine was also held on March 18 (now it's held in mid-May).
Sensoji is said to still enshrine this Buddhist figure (it's a hibutsu - a Buddhist statue forbidden to see), and Asakusa Jinja is a Shinto shrine which worships Hinokuma Hamanari, Hinokuma Takenari and Haji no Nakatomo as divinities. So, on the evening of March 17th, the spirits of the three divinities are put into three mikoshi (portable shrines) at Asakusa Shrine and are carried into the main hall of Sensoji to reunite with the honzon of the temple and stay overnight there. The next morning, the three mikoshi go back to the shrine and the gods are put back to the main building of the shrine.

March 17

On the evening of March 17th, the spirits of the three deities enshrined at the main building of Asakusa Jinja are put on three mikoshi (portable shrines) and carried into the main hall of Sensoji Temple from the front side climbing the stairs. Each mikoshi weighs more than 1 metric ton.

1. Shrine priests put the spirits of the deities into three mikoshi
2. Three mikoshi head to the main hall of Sensoji Temple
3. The mikoshi go into the main hall
4. Temple priests chant sutras

In the evening of March 17, Shinto priests load the spirits of divinities on three mikoshi. There's not much to see during this period. While the priests head for the mikoshi storage, all the lights in the precincts go off. Taking photos is forbidden during this period.

Then the three mikoshi are moved and placed in front of the main hall of Sensoji Temple.

After that, the mikoshi are carried into the main hall one by one.

When all three mikoshi are carried into the hall, sutras are read, though ordinary people cannot see what's going on there since you cannot climb up the stairs.

March 18

After spending a night at the main hall of Sensoji Temple, the three mikoshi head back to Asakusa Shrine.
A binzasara dance and three dragon dances are also performed this day.

1. Shrine priests chant norito prayers and Buddhist priests chant sutras inside the main hall
2. Three mikoshi get out of the main hall
3. A binzasara dance is performed in front of the main hall
4. The mikoshi parade around the main hall once and go back to the shrine
5. Temple monks hold a ritual at Sensoji
6. The mikoshi get into the warehouse
7. Spirits in the mikoshi get unloaded to be put back in the shrine building

The three mikoshi will be placed on the main approach for a while. In the late afternoon, the mikoshi will be restored in the storage, and the spirits of deities loaded on the mikoshi will be put back to the shrine building.

Mikomai dance is perfomed at 1pm.

Dragon dance is performed three times on this day.
This dance was created in 1958 based on a legend that when the statuette of the Sho-Kannon was recovered from the river in 628, a golden dragon descended to earth from the sky.

When March 18 falls on weekend

If March 18th is a weekday, the mikoshi procession after it gets out of the main hall is very short - just circling the main hall once and then goes back to the shrine. But if the day falls on Saturday or Sunday (like 2017 and 2018), then the distance mikoshi proceed gets much longer, parading through the town of Asakusa and then getting back to Sensoji and Asakusa Shrine.

Hikan Inari Jinja Reitaisai

On March 18th, held as well is the Reitaisai (annual grand festival) of Hikan Inari Shrine located in the northeastern corner of Asakusa Shrine's precincts. In this event, beside a ritual, a few performances will be dedicated to this small shrine. This event has nothing to do with the Honzon-jigen-e, though.

The ritual is held on 11am.

The kinds of performances offered to the shrine differ by the year.