As happened centuries later with a handful of Renaissance painters, ancient Greek art tends to be thought of in vague terms of vases, statues and architecture produced "a long (unspecified) time ago." Indeed - a long time has passed between us and ancient Greece, and thinking like this is a good starting point, really. The vases, sculpture and architecture were huge - huge! - innovations, and artists forever afterward owed an enormous debt to the ancient Greeks.
Because so many centuries and different phases encompass "ancient Greek art" what we'll try to do rather briefly, here, is to break it down into some manageable chunks, thus giving each period its due. Sort of like Greek Art giving an acceptance speech at an awards ceremony, in which it thanks all of "the little people" for helping it to become eternally memorable.
What were the different phases of ancient Greek Art?
There were many phases from the 16th century BC, until the Greeks suffered defeat at the hands of the Romans at the Battle of Actium in 31 BC. The phases are roughly as follows:
Mycenaean Art occurred from roughly 1550 to 1200 BC on the Greek mainland. Although the Mycenaean and Greek cultures were two separate entities, they occupied the same lands, successively. The latter learned a few thing from the former, including how to build gates and tombs.
Besides architectural explorations including Cyclopean masonry and "beehive" tombs, the Mycenaeans were awesome goldsmiths and potters. They raised pottery from merely functional to beautifully decorative, and segued right out of the Bronze Age into their own insatiable appetite for gold. (One suspects that that the Mycenaeans were so wealthy they weren't satisfied with a humble alloy.)
Around 1200 and the Homeric fall of Troy, the Mycenaean culture dwindled and died, followed by an artistic phase known both as Sub-Mycenaean and/or the "Dark Ages". This phase, lasting from c. 1100 - 1025 BC, saw a bit of continuity with the previous artistic doings, but no innovation.
From c. 1025 - 900 BC, the Proto-Geometric phase saw pottery beginning to be decorated with simple shapes, black bands and wavy lines. Additionally, both technique in creating, and shapes of pots were being refined.
Geometric Art has been assigned the years of 900 - 700 BC. Its name is utterly descriptive of the art created during this phase. Pottery decoration moved beyond simple shapes to also include animals and humans. Everything, however, was rendered with the use of simple geometric shapes.
Archaic Art, from c. 700 - 480 BC, began with an Orientalizing Phase (735 - 650 BC). In this, elements from other civilizations began to creep into Greek art. The elements were those of the Near East (not exactly what we think of as "Oriental" now, but remember the world was a lot "smaller" in those days).
The Archaic phase is best known for the beginnings of realistic depictions of humans and (no coincidence) monumental stone sculptures. It was during the Archaic that the limestone kouros (male) and kore (female) statues were created - always showing young, nude, smiling persons.
(Interesting aside: The Archaic, and subsequent Classical and Hellenistic periods each contained separate Early, High and Late phases - just like the Italian Renaissance would further on down the road.)
Classical Art (480 - 323 BC) was created during a "golden age", from the time Athens rose to prominence, to Greek expansion and right up until the death of Alexander the Great. It was during this period that human statues became so heroically proportioned. Of course, they were reflective of Greek Humanistic belief in the nobility of man and, perhaps, a desire to look a bit like gods - as well as the invention of metal chisels capable of working marble.
Finally, Hellenistic Art (323 - 31 BC) - quite like Mannerism - went a wee bit over the top. By the time Alexander had died, and things got chaotic in Greece as his empire broke apart, Greek sculptors had mastered carving marble. They were so technically perfect, that they began sculpt impossibly heroic humans. People simply do not look as flawlessly symmetrical or beautiful in real life, as those sculptures - which may explain why the sculptures remain so popular after all these years.
We'll cover each of these phases in turn but, for now, it's important to know that ancient Greek art was mainly comprised of vases, sculpture and architecture, lasted around 1,600 years and covered a number of of different periods.