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. 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.

Why We Think Soriel Wouldn't Work.
Gerson specifically tells you that Asgore and Toriel were at the battle with humanity 1000 years ago. Also, given that Papyrus and his brothers appeared recently (according to the people of Snowdin and often refer to them as 'young skeletons'), it's clear that Sans is not old at all. Therefore the relationship would cause Toriel to be a pedophile.

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 shippers point out that Soriel has no canon proof, usually think Toriel and Sans are friends, and that Toriel and Sans are more than a thousand years apart of age (look at the previous parapragh).

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

Chara Dreemurr
Chara was the adopted child (of unknown gender) of Toriel and Asgore.