Darillium was a planet known for its Singing Towers, a pair of monoliths. The Twelfth Doctor took River Song to see these towers the last time she saw him before she went to the Library and met the Tenth Doctor.
Darillium was located in a galaxy near the Andromeda Galaxy. The planet had several rings around it and a large number of non-spherical moons. Seen from afar, Darillium had a reddish-brown colouration, likely due to having a largely rocky surface. The atmosphere was breathable as well.
It rotated very slowly; in human terms, a night lasted 24 years. Darillium also contained a cave system with a crystal layer. When the wind blew though the caves by the Singing Towers, it would harmonise with the layer and cause a song to be produced, the reason why the monoliths had been named as they were.
In the TARDIS, an older Eleventh Doctor and River Song stumbled upon his younger eleventh self before visiting Darillium, and the two Doctors briefly discussed that it was her last meeting with them. (HOMEVID: Last Night) This conversation made him unwilling to go to Darillium yet and the Doctor cancelled his plans at the last minute. He continued to back out of this date several times.
After the Harmony and Redemption was struck by a meteor storm, it went on a crash-course for Darillium. The Twelfth Doctor and River attempted to stop the crash, but ultimately were forced to flee into the TARDIS. The ship crashed on Darillium, leaving no survivors, save the Doctor and River, who were in the TARDIS during the crash, and Nardole and Ramone, whose heads were later able to take control of Hydroflax's body, which was recovered from the wreckage.
After the crash, the Doctor decided to stop fighting the inevitable and take River on their last date together. The Doctor met a man named Alphonse who had come to the wreckage of the Harmony and Redemption in search of survivors. The Doctor recognised that he was benevolent and, in preparation for his date, gave Alphonse the Halassi Androvar diamond, telling him to redeem the reward and use the money to build a restaurant in front of the Singing Towers. The Doctor then travelled forward in time to after the restaurant was built to make a reservation. Although the restaurant was fully booked up to Christmas Day in four years time, he made a reservation for "table on the balcony with a view of the towers" and travelled to the day of his reservation with River.
Having been knocked out by the force of the crash landing, River eventually awoke and left the TARDIS, entering the restaurant and using a special perfume spray to change clothes. There, she found Nardole and Ramone working at the restaurant and was greeted by the Doctor, who gave her her first present from him: a sonic screwdriver. The Doctor took her to the balcony to see the Singing Towers. River revealed knowledge that this was the last night they spent together though the Doctor refused to confirm it. He eventually told her that night on Darillium was twenty-four years long, giving them plenty of time on their last date. (TV: The Husbands of River Song)
River told the Doctor in the Library that "the last time I saw you — the real you, the future you, I mean — you turned up on my doorstep with a new haircut and a suit. You took me to Darillium to see the Singing Towers. What a night that was. The towers sang and you cried. You wouldn't tell me why, but I suppose you knew it was time. My time. Time to come to the Library. You even gave me your screwdriver. That should have been a clue." (TV: Forest of the Dead)
Out of fear of being alone once River was gone, the Doctor removed Nardole's head from Hydroflax's body and rebuilt Nardole's original body. Nardole later commented that it was a hell of a twenty-four years the Doctor and River spent on Darillium. (TV: The Return of Doctor Mysterio) In addition, Missy learned about the Doctor's time on Darillium from the Daleks and mentioned it to him before her scheduled execution. (TV: Extremis)