Role in the Film
Appearing toward the film's end, she and her children were out for a swim when they overheard someone crying nearby. The sound led them to a lone little cygnet who mistakenly believed himself to be a duckling due to being hatched in a duck's nest.
Because he was so different from their other ducklings, the ducks had cast him out of their family and his failure to find acceptance anywhere else because of his "ugliness" had crushed his hope of ever being loved. After her cygnets introduced themselves and convinced him to come and play, she called out to them, but the Ugly Duckling's fear of further rejection made him hesitant to even approach her.
However, Mother Swan pulled him close (despite his initial resistance) and lovingly embraced him to prove that she didn't consider him ugly at all and that she's willing to take him in due to his plight. Realizing this, the Ugly Duckling happily embraces the Mother Swan, accepting her and the other cygnets as his family.
As the short ended, the Mother Swan was last seen leading all of her cygnets home (including the Ugly Duckling). It was never specified in the film if she was his real mother who somehow lost the egg or an adoptive mother, but either way, she gave the Ugly Duckling what he needed the most: a loving family.