Brief Generic "Rules"

These are pretty much the most important things, roughly in order, that you need to have if you want a successful fanfic.

Obviously, the most important of these is the actual story: make it good. Make it interesting. If it's (randomly scanning mind for example) a Mario fanfic, don't have Peach get kidnapped again unless you can make up for it somehow with really good, crisp writing.
(Remember, there are exceptions to all of these "rules".)

The second most important thing is to have your sentences flow. If your story is like this:

no phoenix said. jack daniel must have been the killer because
all of the evidence points to it what do you think edgeworth he asked
edgeworth?. edgeworth just said you're stupid wright.

--then you need to go back and do some exercises to get your talent to express itself better.

The almost-second most important thing (meaning these two are really practically close enough for them to be considered a tie) is good characterization. This is something that non-fanfic writers rarely have to worry about as much. If they mess up, it's unlikely anyone will call them on it. But the people reading fanfics know about as much (maybe even more) about the "universe" the story's being written in as you do.
Here's the thing, though: Look at that as a blessing. So there's a bunch of nitpicky fans? Great! Send 'em in! Even though some of them may go about it the wrong way (if they're annoying, you can always not read their remarks), they're helping you, and most of them really do have your best interests at heart when they review.

The third thing is grammar, which is, again, connected to "flow"--bad grammar is painful to read, as you can see here (I had to almost literally kick myself in order to write this):

There said man, who wasn't of course the guy they'd ben searched for. "Maybe one day hel'l be come back here" the texan rangers said, messing with his tall and brown and stiff hat that his head was perched on by.

Man, now I feel like a failure because that is definitely not a good example--there are instances of grammar foul-ups so astonishing out there in the universe that they even make Chinglish look tame. I guess the problem with mine is that I tried.
Now, grammar is (somewhat) subjective. For example, if you write in a conversational style, it is perfectly all right to begin sentences with "and" sometimes, because that's how people talk. (More on the flexibility of certain rules in a future post.)
Obviously not all characters have to have good grammar, but choosing those kinds of things is a discussion for another time.. ^_^

The fourth most important thing is punctuation. I cannot tell you how important those little marks are. To paraphrase someone who came before me, punctuation is like a series of traffic signals: they are provided so that readers know when to stop, slow down, and keep driving. They are meant to give a bit of order, pacing, and style to the act of reading.
Using punctuation shows that you care about the reader: in today's high-paced society, to stop and use those little tiny marks shows that your readers mean a lot to you.

And then of course we have spelling. Never fall victim to the delusion that Microsoft Word has got your back: it fails to look at things like context. For example, Word may not see anything wrong with this sentence:

The body lay on the ground, the bear skin devoid of marks.

But there is something wrong with the sentence: the author meant to use the word "bare", meaning without a covering, but ended up (maybe even through a typo) with "bear"--which, technically, is spelled correctly.
Even if Word was better at sensing context, an author would be screwed with a sentence like the one above. It could be read as meaning "the [skin of a bear] was devoid of marks..." and still be a perfectly legitimate sentence--to a computer.

So, in summary: you can't get anywhere if your idea's not good. But you won't do justice to a good idea if you don't bother to execute it properly.