The history of the language that you speak is carved on its very words that, spoken and sung by millions of people over thousands of years, bring the history of our nation to us.
Ancient Greeks spoke in different ways from place to place. That means that the language was taking different forms from place to place, forms which are known as dialects. In Sparta they had the Doric dialect (δωρική διάλεκτος), in Athens they had the Attic dialect (αττική διάλεκτος) and other dialects in other places. This, however, didn't prevent them from communicating and understanding each other without great difficulties, because the differences between the dialects were not very significant. Their language was virtually the same. Take the Modern Greek word μητέρα (mother) for instance. In Athens, it was μήτηρ, whereas in Sparta it was μάτηρ. The Athenian article τήν was τάν in Sparta. As you can see, the differences were not this important so as to make a different language, but only a different dialect.
The common language (κοινή γλώσσα), however, was developed much later, after the Classical Age, in the years between 300 B.C. and 300 A.D. This language, that gradually became the common language of all Greeks, came from the Attic dialect and in particular the one spoken in Athens between 500 and 400 B.C., namely the time that Athens had reached the zenith of their glory. But in order for the Attic dialect to become a common language it had to borrow words from other dialects as well. This happened at the time that Alexander the Great was conquering Middle East, so this common language became (with time) a language of the Middle Easterners as well. It was spoken and written everywhere: Syria, Asia Minor (modern Turkey), Persia (modern Iran), Egypt etc. It had become in a way the international language of that era.
This language gradually (and because it became widespread) lost its old form and developed some new features that foretold the language that is spoken today. The language of the 3rd century A.D. shares many elements with modern Greek. Instead of the ancient words ύει, ερυθρός, ιχθύς (it rains, red, fish) they said βρέχει, κόκκινος, (ο)ψάρι(ον), pretty much as we do today. We can find many similar examples in the New Testament, which was originally written in the common Greek language of that time. That is why it is argued that the common language of the Christ era is a milestone in the history of the Greek language and was the first form of Modern Greek.
More changes came in the Byzantine era. Thus we can find many words in their contemporary form παιδί(ον), πατέρας, πόλη, ας, να (child, father, city, let's, to) and many more. Later, when the Byzantine empire diminishes, the spoken language is almost no different than the modern language, Modern Greek, the way we know it from the folk songs.
Modern Greek language has unique virtues : expressibility, flexibility, composing power, productivity, which means that it composes and produces words according to the needs of the speaker.
Greek language, as you saw, has a long history. It has the longest history among the european languages. Scriptures in Greek have been discovered that date back to the second millenium B.C. and literature works more than 2500 years old. Works that will never become outdated have been expressed through this language: the ancient literature, the New Testament, the byzantine and modern Greek literature. This is the language that our people have used to raise their own monuments : the fairytales, the proverbs, their traditions, our folk songs.
It is a treasure. But this linguistic treasure of ours we have to conquer. And in order to achieve this we need some tools. One such tool is the Grammar, this book that with love we offer to the Greek student and the Greek nation in general.