I went with Aranea Ienith, a Dark Elf from the Shrine of Azura, to the Labyrinthian to finish some quest. When we killed the boss, I told her we should part ways, because I didn't need her anymore and I wanted to go to Whiterun to sell some stuff.
I went with her until she passed through the first door. After she went through, she appeared on the other side and just stood there, in place. She wouldn't move. I thought she'd move eventually and continued further.
After selling some stuff in Whiterun, I went to Shrine of Azura to see how she was doing, but I didn't find her there. Could she be in Labyrinthian where I last saw her standing?
I also heard you could wait for about 3 days and she'll be back at her original place. Or is that only if you tell her to wait?

I'd go back to Labyrinthian first and if she's not there go home and wait a full 30 days. My Daedra worshipping Dunmer uses her a lot, and takes her back to the shrine on foot whenever he needs to do quests solo.

While you're waiting, here's the pair of them getting a piece of a Daedric artefact together on master diff.

http://www.youtube.com/watch?v=J3mwrQrBl7g

If you're on a PC, the command

player.moveto 28ad1

Will take you straight to her. Otherwise you probably have to wait some days.