Syntax is the linguistic study of how words combine to form sentences. (To linguists, "syntax" is a subset of "grammar", and grammar also includes things like the study of valid word forms.)
Generative grammar and loglangs
There are many broad theories of how syntax arises, rooted in philosophical questions. Why are humans so good at language? How do humans acquire language so quickly, and why do they make some kinds of mistakes but not others?
One such theory is generativism. It posits that the human brain has an innate faculty for grammar, which places certain restrictions on the parameters of human languages. We can imagine a universal grammar "generated" by this innate faculty. There have been decades of efforts to describe this universal grammar, and show that human languages all over the world adhere to it in some sense. For example, human languages tend to have noun phrases and verb phrases.
This theory is not without criticism: there's little neurological evidence for an innate "language device". However, generativism has also been very successful in explaining natural language syntax and semantics at many levels. If we can describe Toaq's syntax in these same terms, we can be certain that it is a human language, rather than merely a way to "speak out loud" an unnatural mathematical or logical structure.
Hoemaı's attempts to describe Lojban syntax in the framework of generativism were unsuccessful, whereas efforts to describe Toaq with the same linguistic tools are working out (and Toaq is evolving with this goal in mind). This current description of Toaq syntax is influenced by "X-bar theory" and the "Minimalist program" — sub-theories of generativism with certain ideas about syntactic structures. The forefront of linguistic knowledge has progressed a bit beyond these theories, but they are still very adequate frameworks to serve as points of reference. (As conlangers, we can "cheat" a little and design Toaq so as to not bump into the flaws of these more fleshed-out systems.)
Toaq being a loglang means that we can unambiguously parse sentences into syntax trees. Zugaı is a piece of software that performs this transformation.
(TODO: touch on criticisms of PEG/camxes)
Movement
Generativist syntacticians say that sentences have a "deep structure" that adheres to universal grammar, but various language-specific constraints transform this into the "surface structure" when the sentence gets actually realized. The most important such transformation is syntactic movement.
For example, English has something called wh-movement: when we turn a sentence like "Mary wants Bill to dance" into a wh-question, we say "Who does Mary want () to dance?".
The generative explanation for this is that the question has a deep structure like "Mary wants who to dance?", and then for pragmatic reasons, the question word moves to the front of the sentence and gets supported by "does". There is a trace marked by () in the spot where "who" moved from.
There is good evidence for wh-movement. English speakers tend to agree that we can't contract the sentence to "Who does Mary wanna dance?" — we can imagine the "who"-trace between "want to" is blocking the contraction.
Note that the claim is not that the deep-structure sentence first forms in the speaker's mind, and is then rearranged into surface-structure. The temporal "before and after" perspective on movement is only a useful metaphor for a language's grammar rules.
Movement in Toaq
(FAQ: Why do Toaq sentences need so much movement? Why the "underlying SVO" analysis? Why isn't Toaq simply designed, for simplicity's sake, to have deep structure = surface structure?)
(Could touch on history, aesthetics, comparison to P(x,y,z), analysis of real-life VSO langs, "FP", genitival serials, 
.)