The use of columns as vertical supports was common long before the Greeks. The ancient Egyptians, for example, built structures using columns decorated with hieroglyphs and other motifs — the Temple of Karnak, with its enormous pillars, was built around the 15th century BC. Around 700 BC, the Dorians and the Ionians (two of the main ethnic categories of ancient Greeks) began constructing their columns in specific ways. A couple centuries later, the Corinthian column emerged, supposedly (according to Vitruvius) inspired by an overturned votive basket left on the grave of a young girl, where it became overgrown with acanthus leaves.
The watchword of the ancients was Proportion — it was this concept in art and architecture that inspired the Renaissance (Leonardo's famous Vitruvian man was based on a section of Vitruvius where he discussed geometry and human proportions.) The different types of columns, therefore, each had different rough proportions, different ratios between height and width, that contributed to the sense of elegance or sturdiness, and that helped define the order.
Vitruvius identified three main ancient orders as Doric, Ionic and Corinthian, but Renaissance architects noticed more variation in the examples still extant from ancient times, and added the Tuscan and Composite orders to the list. Each order is defined not only by its proportion and decoration on the column itself, but also how the column interacts with its entablature and base. For our purposes, though, let's take a look at how to quickly identify and define the major column types.
Tuscan (images 2 & 3): Serlio believed the Tuscan order to be the earliest and most rustic of the orders. Never fluted, it supposedly sits on a round plinth, while the others sit on squares. Stocky and sturdy, it was described by Vitruvius of having a diameter ratio of 6:1, while Renaissance architects like Vignola expanded it to 7 or 7.5:1 to give it a little more grace. More recently, its simplicity made it popular in vernacular Georgian architecture in America and England.
Doric (images 4 & 5): The Doric column is similar to the Tuscan order in its squat sturdiness. In fact, both have historically been perceived as 'masculine' and even 'soldierly,' devoid of any ornament other than the occasional fluting. Doric columns typically sit flat on the floor, with no base or pedestal. The Doric sobriety and simplicity has inspired architects starting in the mid-18th century, but the mode was mostly ignored during the Renaissance. Vitruvius determined that the proper Doric proportion was 7:1.
Ionic (images 6 & 7): While the Tuscan and Doric orders were considered masculine, the Ionic order was seen as more elegant and feminine, with a graceful proportion of about 8:1. The main characteristic of an Ionic column is its capital, composed of four scrolling volutes with typically egg-and-dart decoration between them. The column shaft is often fluted, and it rests on a stepped base.
Corinthian (images 8 & 9): The most ornate of the ancient orders, the Corinthian column first appeared in 5th century BC Greece, but was used infrequently, only becoming common during the Roman Empire. Considered feminine like the Ionic order, the Corinthian is still more slender and elegant. Its capital is decorated with scrolling leaves — typically acanthus — and usually incorporates loosely scrolled volutes, as well. The Composite order (image 10) is essentially the same as the Corinthian, but more of a distinctive hybrid with the Ionic capital with its tightly scrolled volutes. It developed during the Roman Empire, and has remained a very popular variation on the Corinthian. Both the Corinthian and the Composite typically have an elegant 10:1 proportion.
Sources: 1 Wikimedia Commons; 2 Artnet.com; 3 Desire to Inspire; 4 Wikimedia Commons; 5 Design*Sponge; 6 travels.co.ua; 7 Dwellers Without Decorators; 8 Wikipedia; 9 Habitually Chic; 10 Artnet.com.