When, Where and Why
Sanja Matsuri is an annual festival that takes place during the the third weekend of May at Asakusa Shrine in the Asakusa district of Tokyo. This year, the festival takes place from May 19th - May 21st. This festival attracts about 1.5 - 2 million people, so be prepared to be a little squished.
The festival is celebrated in honor of the three founders of Sensoji Temple: Hinokuma Hamanari, Hinokuma Takenari, and Hajino Nakatomo. The temple is next door to the shrine where the founders are honored.

Day 1
Suggested Arrival: 11AM
The festival has a bit of a religious element to it since it's celebrated at a temple. The celebration starts off with the Daigyoretsu Parade, where priests, geisha, musicians and dancers walk through the streets wearing traditional Edo period costumes.

The parade goes from Yanagi Dori to Sensoji Temple and Asakusa Shrine where a Shinto ceremony is performed. The head priest moves the enshrined kami representing the three founders from Asakusa Shrine to the three portable Shrines (mikoshi) that will be paraded through the streets throughout the weekend.

After the ceremony mikoshi from Asakusa's local neighborhoods are paraded through the streets, this is followed with traditional music played on taiko drums and flutes.
Day 2
Suggested Arrival: 11AM
If this is your first Japanese festival you may be a bit exhausted, but this party isn't slowing down! The 2nd day is when nearly 100 mikoshi from the surrounding 44 neighborhoods. People carry their mikoshi from Nakamise-dori to Hozomon, where they pray to Kannon, the Goddess of Mercy. The mikoshi are then carried to Asakusa Shrine to be blessed and purified by Shinto priests for the coming year. When the ceremony is completed the mikoshi are carried back to their respective neighborhoods and paraded around to spread good fortune.
With so many mikoshi being moved around it can get a bit overwhelming, but don't worry the event is all in good fun and no one gets hurt.

Day 3
Suggested Arrival: 5AM
Day 3 is definitely the earliest time the festival starts. The final day begins with people from all 44 neighborhoods gathering in matching costumes around Asakusa Shrine. They compete to carry one of the three main mikoshi that represent the founders of the temple. Because of safety concerns, spectators aren't allowed to enter through Sensoji's gates.
After about 2 hours the mikoshi are separated and carried throughout the district. The mikoshi eventually make it through all 44 neighborhoods and returns back to the shrine in one piece.

Other Attractions
Of course there are plenty of other things to do besides watch people parade through the streets. There are plenty of shops and food stalls that you can visit around Nakamise.
You can also buy tickets to one of the best Geisha shows! The Geisha perform on Saturday afternoon from around 1-3PM on the second floor of the Asakusa Kenban, tickets must be purchased, and they sell out fast, so be early!
If you want to plan out your day you can print out the festival's event schedule, these are usually posted on the festival's website (I'll also list sources below).
This is one of Japan's most famous festival's, so try to make it to at least one day. This is a great time to learn more about Japanese culture while also enjoying the arrival of Spring!
Sources:
1. http://www.japan-guide.com/e/e3063.html
2. http://www.japan-talk.com/jt/new/sanja-matsuri-Tokyos-biggest-and-wildest-festival
3. https://en.wikipedia.org/wiki/Sanja_Matsuri#Day_by_day