Asgoriel

Asgoriel is the het between Asgore and Toriel from the Undertale and Deltarune fandoms.

Undertale
Toriel and Asgore were married and the royals of their kingdom. They had one child called Asriel and an adoptive child called whatever the player chooses as the human's name (Chara). However, when Chara died, Asriel absorbed their soul and took them to the humans' world. They were severely injured by the humans. They could easily have killed all of them, but they didn't fight back. They staggered home and died in front of Asgore and Toriel. Usually kind, patient and peaceful, Asgore was filled with grief which later turned to rage. He wanted to collect seven human SOULs in order to break the barrier and to declare war against humans. Toriel was disgusted by this and left him. After that, Asgore's anger melted and was replaced by sorrow.

At the end of Undertale (Pacifist Route), Asgore still misses Toriel. However, Toriel doesn't return the feelings and even refuses to be friends with Asgore.

At the end of the Pacifist Route, however, they must have reconciled to some degree, since he now cuts hedges in front of Toriel's school.

Deltarune
Toriel and Asgore were together at some point and had Asriel. They also adopted a human child called Kris.

Asgore longs after his past relationship with Toriel, and still sends her bouquets of flowers (which she throws away). He keeps pictures of their wedding on his fridge.

Children
Asriel Dreemurr Asriel was the son of Toriel and Asgore. His name is the combination of Toriel and Asgore's names. He appears in Undertale and is mentioned in Deltarune. 

Chara Chara was possibly the adopted child of Toriel and Asgore. Chara appears exclusively in Undertale. 

Kris Kris is Toriel and Asgore's child in Deltarune. 

Fanon
Most Asgoriel shippers like to think that Toriel and Asgore get back together after the events of Undertale, while others only ship Asgoriel during the events before Undertale.

Its biggest rival ship is Soriel. Some Asgoriel shippers think that Soriel has no canon proof, and usually think Toriel and Sans are friends.

The flowers Asgore has preserved are thought to be Toriel and him's wedding flowers.

On AO3, Asgoriel is the most written ship for both characters and the 9th most written ship in the Undertale fandom. It is also the second most written Undertale ship not involving Sans or Papyrus.

Fandom
FAN FICTION
 * Asgore/Toriel on FanFiction.Net
 * Asgore/Toriel on FanFiction.Net