By syntax I mean primarily sentence logic. The two most frequent problems here are misplaced clauses and faulty agreement. But the message is a more general one: make sure your sentences mean what you think they mean.
A perennial problem in writing is the misplaced or floating clause, especially the introductory clause. Consider this sentence. "As an economist, costs and benefits are important concerns to me." It seems to make sense, and some people would probably say it's OK; but it sounds illogical to me. The clause "as an economist" is obviously intended to refer to the speaker, whereas by its placement it actually modifies "costs and benefits." Costs and benefits are not an economist. There are of course far more egregious examples of this sort of thing, like this one. I like the old rule: make sure that the word immediately after an introductory clause is always the word you want the clause to modify.
The problem of misplaced clauses goes beyond the problem of agreement. Because English is not an inflected language (it doesn't have lots of case endings like, say, Latin), word placement is critical for conveying meaning. Indeed, sometimes things can go dangerously wrong when clauses get misplaced.
Another big problem of misplacement involves making clear what your pronouns refer to. Here too things can go terribly wrong.
Agreement in number.
Consider these sentences. "Each of the billions and billions of stars is a sun like our own. None of them are made of green cheese." Or should it be: "None of them is made of green cheese"? The old rule is that words like none, any, and each are singular and should take singular verbs. Students of formal structural grammar say that this is not so: in the sentence above, a native speaker construes "none of them" as the subject, not just "none," and construes it as plural. I'm old fashioned: I would still use a singular verb in the case above.
Placement of "only."
"I only have eyes for you," says the song. What this literally means is that I have eyes -- and no other bodily parts -- for you. Presumably, the singer really intends something like "I have eyes only for you." While it has become conventional in spoken English to let "only" sit near the verb even if it modifies some other part of the sentence, it still doesn't sound right to me in written English. Put the word near what it really modifies.