In a word, charisma.
I want to read something that draws me in and makes me want more. For that to happen, there are a few necessities.
*Good quality writing - not necessarily professional level, but good enough that the mistakes don't make it annoying to read. Spelling mistakes are bad.
*Compelling and engaging characters. As always, quality is a must. I'd rather read a well-done cliche than a badly-written "unique" character. These characters should feel like real people, not a hodgepodge of "kewl" concepts.
*Meaningful character relationships. Star Trek is all about character relationships, and the greatest Trek characters are largely defined by their relationships with the other characters around them.
*An interesting story. I want the story to go somewhere, and I want it to take me along for the ride. The story needs to flow in a way that I find appealing.
The single largest thing I look for in a Star Trek fanfic...
*Star Trek. Well, obviously. But it needs to feel like Star Trek! I don't want a darker, grittier re-imagined universe full of death and despair. I don't want the main characters to be ruthless, amoral, backstabbing and manipulative (unless it's a Garak fanfic.) I want actual Star Trek in my Star Trek.
About the other stuff...
I like romance. I really do. But a romance won't keep me interested long-term, there needs to be more to the story. I 'ship like crazy, but it's not all I want to read about. Believe it or not, there are people who aren't interested in romance! Imagine that!
One thing romance-wise that bugs me is when two characters are written as if there's something between them and the author seems to build it up as an important part of the story...but abruptly forgets about it near the end. Nuh-uh. If you're gonna make me hope the boy gets the girl, the boy had better get the girl. If you don't want to include that, I'm totally fine with it. But don't begin something and simply drop it - of course, that applies to other things too.
Fanfic focusing on canon characters tends to disappoint me, simply because it's never quite as good as actual Trek and some character voices are just too hard to capture. For that reason, I prefer original characters. I also want to see more of the Trek 'verse than we've seen onscreen. I want to meet new characters.
The length of the story doesn't really matter to me, but I do enjoy reading something with an actual story arc. But not too long, or I get tired of waiting for the resolution. As long as something actually happens (they beat the bad guys, they find the alien treasure, they discover the truth about the mysterious artifact...) and the story doesn't become boring, I like it.