Moe-chan is a mechanical doll, made over a hundred years ago. She appears in episode 20 or so of the anime.
In episode 20 of the anime, it is revealed that Keisuke Urashima a made a promise to Moe-chan to fix her up. He never kept that promise.
She sat in the Hinata-sou's basement during this time until Keitaro Urashima discovered her in the basement. Keitaro convinced Kaolla Su to fix her up and she wound up being glad that the promise was kept.
At the end of the episode, she ran down and stopped working - perhaps out of energy. Keitaro put her back in the basement and she was never seen again until Love Hina Again episode 3.
In Love Hina Again episode 3, she appears to Naru Narusegawa in a dream and tells her where the ring is. After the explosion, she appears for the last time in the series smiling at Naru.