The term ‘order’ means the basic style of a classical temple. There were five standard orders in the classical world, with three (Corinthian, Tuscan, composite) derived from two fundamentals (Doric, Ionic). We will also consider the Egyptian column.
First, on columns. Their first use was as a single central roof support for small buildings during the Bronze age (3000BC-1000BC). More sophisticated columns appeared in the Egyptian, Assyrian and Minoan civilisations. While the formers used stone, the Minoans often used whole tree trunks, turned upside down, stood on a base, topped by a round capital, and painted. As a result, many of them have not survived, leaving only their stone bases.
Ancient Greece marked the transition to stone columns for Hellenic civilisations, but the transition was not clear-cut. While some were carved from one piece, many were instead constructed from multiple drums, fitted together with wooden or metal pegs. Such columns were very resistant to seismic activity, requiring strong storms or intentional weakening to topple. Despite these advantages, the Romans preferred single shafts.
An interesting note - due to the distortion of geometry due to a ground-based perspective, many columns actually leaned slightly inwards as they rose to counteract the effect of divergence. The corner columns were also slightly fatter, and all columns bulged slightly in the middle (entasis) as well. The most famous example of these techniques is the Parthenon.
Columns became so important that they became major stylistic features by themselves, such as the bull-man columns of Persepolis, and the caryatids of the Erechtheion.
Free-standing columns also began to appear, such as the Ionic Naxian Sphinx in Delphi, and Trajan’s Column in Rome, featuring the continuous helical frieze of his exploits. This heavily mimics the tradition of Egyptian obelisks, which were developed much earlier, in around 3000BC. However, there are significant differences. Notably, while columns are round, obelisks (Greek for ‘spit’, tekhnu in Egyptian) are flat, and topped with a pyramidal shape. Furthermore, the primary use of columns was for construction, with decorative or commemorative free-standing columns arising only later. Obelisks instead were always commemorative or religious, believed to be a way to preserve the spirit of a king or emperor while the soul fled to the afterlife.
This is not to say the Egyptians did not have columns - the very much did. Rather, they differentiated the purposes of columns and obelisks from the very start.
Now, on the orders of the ancient world.
The two fundamental orders were both Greek: the Doric and the Ionic.
The Doric column has no base (no plinth, rises straight from the ground). It is fluted, but with no sharp edges (fillets). The capital of the column is square, with no decorations. The Doric architrave (the lower level of the horizontal slab supported by the columns, see the picture) is plain, with a frieze (next level) decorated with alternative triglyphs (3 V-shaped notches) and metopes (the space between two triglyphs, sometimes decorated, think the Parthenon metopes).
The Ionic column instead has a base (the standard one is a thick bead, or torus, see the bottom of the columns in the picture). The shaft is fluted, with sharp edges (fillets). Finally, the capital has a pair of connected spiral decorations (volutes). The architrave (lowest level of slab) is divided into three steps. Then the frieze is plain, with perhaps some continuous sculpture. Finally, the conice (uppermost level) has small evenly spaced blocks of moulding called ‘dentils’, because they look like teeth.
Now we come to the derivative orders. The first is Corinthian, developed from the Ionic. The main difference is the capital, which is a lot more flowery. It has miniature volutes (the swirly bits), and staggered rows of acanthus leaves.
The Romans adapted all of the above to suit their own tastes, but also developed two of their own styles. The first of these is Tuscan. The Tuscan order is basically Doric, but with the column fluting and triglyphs removed, resulting in very plain and austere columns. They also added a plinth to the base in most cases.
Finally, the composite order, which is essentially the Corinthian order, but the capitals were a hybrid of Corinthian and Ionic capitals, with both larger volutes and the acanthus leaves.
The Egyptian column featured a base, and rather than fluting, featured sculpted decorations of animals and leaves. Papyrus reed featured heavily. Also, rather than the more regular style of greek and roman shafts, their shafts were more irregular, sometimes featuring three or more horizontal sections.