"syntax" refers to the

Simple sentence: Subject-verb structure ("The girl ran.")

Compound sentence: Subject-verb-object-conjunction-subject-verb structure ("The girl ran the marathon, and her cousin did, too.")

Complex sentence: Dependent clause-subject-verb-object structure ("Although they were tired after the marathon, the cousins decided to go to a celebration at the park.")

Compound-complex sentence: Four clauses, dependent and independent structures ("Although they weren't fond of crowds, this was different, they decided, because of the common goal that had brought everyone together.")

Syntax Variations and Distinctions
Syntax has changed some over the development of English through the centuries. "The proverb Whoever loved that loved not at first sight? indicates that English negatives could once be placed after main verbs" (Aitchison, 2001). And not all people speak English in exactly the same way. Social dialects learned by people with common backgrounds—such as a social class, profession, age group, or ethnic group—also may influence the speakers' syntax. Think of the differences between teenagers' slang and more fluid word order and grammar vs. research scientists' technical vocabulary and manner of speaking to each other. Social dialects are also called "social varieties." 
Beyond Syntax
Following proper syntax doesn't guarantee that a sentence will have meaning, though. Linguist Noam Chomsky created the sentence "Colorless green ideas sleep furiously," which is syntactically and grammatically correct because it has the words in the correct order and verbs that agree with subjects, but it's still nonsense. With it, Chomsky showed that rules governing syntax are distinct from meanings that words convey.
The distinction between grammar and syntax has been somewhat disrupted by recent research in lexicogrammar, which takes the words into account in grammar rules: For example, some verbs (transitive ones, that perform an action on something) always take direct objects.1 A transitive (action) verb example:

• 
  "She removed the index card from the old recipe box."
  

The verb is "removed" and the object is "index card." Another example includes a transitive phrasal verb:

• 
  "Please look over my report before I turn it in."
  

"Look over" is the phrasal verb and "report" is the direct object. To be a complete thought, you need to include what's being looked over. Thus, it has to have a direct object.
Additional References

• 
  Aitchison, Jean. Language Change: Progress or Decay? Cambridge University, 2001.
  
• 
  Burgess, Alan. Enderby Outside. Heinemann, 1968.
  
• 
  Chomsky, Noam. The Logical Structure of Linguistic Theory. University of Chicago, 1985.
  
• 
  Copeland, Douglas. Generation A: A Novel. Scribner, 2009.
  
• 
  Miller, Jim. An Introduction to English Syntax. Edinburgh University, 2008.