From [ACOS: Glossary, 675]:

"Juilin Sandar - A thief-catcher from Tear. A man in love with perhaps the very last woman he would ever have thought he could be."

So, who is it? People have suggested an Aes Sedai, Nynaeve, Birgitte, etc. However, there is only one idea backed by any demonstration of tender feelings on Juilin's part: ex-Panarch Amathera of Tarabon:

We know from [ACOS: 14, White Plumes, 273], as well as other places, that Juilin does not like nobles, so a high noble of some sort might seem to be the last person he would think he'd fall in love with.

How about some quotes?

• [ACOS: 14, White Plumes, 274] "Tarabon must be terrible now for a woman not used to taking care of herself," Juilin murmured.
• Going back to the group's trip away from Tarabon [TFOH: 9, A Signal, 146], we have:

  Nynaeve: "Amathera was difficult, but I do not wish her any harm. Do you?"

  Juilin: "A pretty woman, especially in one of those Taraboner serving girl's dresses, with a pretty smile. I thought she...." (shuts up when Elayne shoots him a dirty look.)

This is all confirmed in WH. Juilin convinced Thera/Amathera to run away with Mat and Co., and she joined their party in [WH: 31, What the Aelfinn Said, 586].