Asgoriel

Asgoriel is the relationship between Toriel and Asgore from the Undertale fandom.

Canon
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 and Asriel took them to the humans' world and was later attacked by the humans, Toriel and Asgore had a disagreement. Asgore wanted to collect seven human SOULs in order to break the barrier and declare war against humans. Toriel didn't accept his so she left Asgore.

At the end of Undertale (Pacifist Route), Asgore still has feelings for 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.

Fanon
Most Asgoriel shippers like to think that Toriel and Asgore get back together after the events of Undertale. There are also some people who only ship Asgore during the events before Undertale.

Its biggest rival ship is Soriel. Asgoriel fans point out that Soriel has no canon proof and usually think Toriel and Sans are friends.

Asriel Dreemurr
Asriel is the son of Toriel and Asgore. His name is the combination of Toriel and Asgore's names.