This is a good question and this word is often times defined so ambiguously. Syntax is the study of the principles that govern phrases, clauses, and sentence structure. Each language in the world has rules for sentence structure. Syntax studies these structures and communicates these structures to people.
There is also a pretty decent overlap with grammar. So, if you study grammar and know the rules of grammar well, then you can be said to have a good handle on syntax.
Finally, it is important to study the rules of syntax, because being able to write well and speak well will be one of the ways in which you can get ahead in society. This is also a mark of an educated person.