What Is Meant By Syntax?

Syntax is another name for word order - the way that words are arranged in a sentence and their relation to each other.
Syntax affects meaning - without it, we would speak in jumbled sentences (i.e. it's because of the rules of syntax that we would say "The black dog is chasing the green ball" rather than "ball is dog black green the chasing" or any other arrangement we feel like.
This doesn't mean that syntax is fixed for all time, only that there must be general agrement on the rules if language is to function. In fact the syntax of English (and of many other languages) has changed over time. Middle English word order was quite different; a sentence like "with him there was dwelling a poor scholar" would be "a poor scholar was living with him" today.
Traces of the older syntax can still be found in expressions like "Neither am I" and "never have I been so surprised, but in general, modern English follows the basic syntactical rule of subject-verb-object ("the cat sat on the mat," not "on the mat sat the cat.")