She is Alice's grandmother, and she helped Darkrai when it was injured. She was the first human being to show it any kindness. Darkrai believes that Alice is Alicia and that is why it was so protective of her.
Alicia taught Alice the song "Oración" meaning "Prayer," which has the ability to calm any Pokémon. There was also a special song disc that could be played on the Space-Time tower's chimes. It was in turn used to stop Dialga and Palkia's fighting.
A sculpture of Alicia created by Godey is in the middle of the Space-Time towers. She is playing her song while many Pokémon sit around her and listen.
(Sources: Bulbapedia)