Understanding/interpretation of depression from in classical Greece/Rome and in 18th century are very similar.

In the 18th century, Depression was known as melancholia. According to a write up published on the Stockton University website:

"There were two types of melancholy: natural and unnatural. The first was a condition considered to be brought on primarily by a black bile produced in the human body that could be dried up over time. This was the result of certain foods that would cause such bile to manifest. Strong wines were thought to act in this way...

The second type of melancholy, described as unnatural, was the result of a corruption from demons and spirits. "

http://loki.stockton.edu/~kinsellt/projects/pom/storyReader$7.html

According to Rashmi Nemade, Ph.D., Natalie Staats Reiss, Ph.D., and Mark Dombeck, Ph.D. in their report published here:

http://www.gracepointwellness.org/5-depression/article/12995-historical-understandings-of-depression

"Ancient Greeks and Romans were divided in their thinking about the causes of melancholia. Literature of the time was filled with references to mental illness caused by spirits or demons. In the 5th century B.C., the Greek historian Herodotus wrote about a king who was driven mad by evil spirits. The early Babylonian, Chinese, and Egyptian civilizations also viewed mental illness as a form of demonic possession...

Hippocrates, a Greek physician, suggested that personality traits and mental illnesses were related to balanced or imbalanced body fluids called humours. There were four of these humours: yellow bile, black bile, phlegm and blood."

In both classical Rome/Greece and in the 18th century, depression was understood to be an imbalance of the humors and/or caused by some sort of supernatural possession.

That being said, it is completely possible that during the Middle Ages, beliefs were different from the era before the Middle Ages and after. So I am only answering part of the question here, namely, what depression was perceived as in the "Classical" era.

While it's a little out of the time period you're asking for, I can give you an example of how depression was understood from the early Renaissance. Marsilio Ficino (1433-1499) was a prominent neo-platonist, occultist, astrologer, and humanist philosopher in fifteenth-century Florence. In one of his books, he argued that the planets had an affect on people's character and dispositions, and by associating yourself with certain objects associated with a particular planet you could bring about a change in your character. For example, Ficino felt that he had a saturnine disposition – that is, he was too much affected by the planet Saturn, the planet associated with weight, heaviness, and depression. Different objects were associated with Saturn, including most prominently the metal lead, though you can see others in this Dürer engraving, where the angel's melancholy is accompanied by both a giant block of lead and a magic square invoking lead. Indeed, the links between planets and behaviours are where we get a number of adjectives: someone happy is called jovial, while another person might have a mercurial personality. Amusingly, the adjectival form of Venus is found only in the phrase 'venereal disease'. Ficino argued that one could alter one's disposition by associating oneself with the appropriate objects – gold and roosters for the sun, for example. He also performed what he called Orphic Hymns, also designed to realign his character.

And this is the part that makes me nervous: I'm currently on vacation and am away from all of my sources. However, I'm happy to amend this section in about two weeks' time.

At a high level, the idea of Humorism (Which originated well before BCE and lasted until the 19th century) saw depressed people, at least in part, as Melancholic. Depressed, brooding, focusing on all that's wrong, etc.

While Melancholia was sometimes a conscious affectation, it was also seen to be the result of too much "black bile" in the system. The activities, age and diet of people were thought to influence their humors and the biles that influenced them.

Avoiding foods and activities that were "cold and dry" would be a way to curtail the excess of black bile, as "cold and dry" were seen as the nature of black bile.

It's also possible that a person's spleen would become suspect, as this was thought to be the organ responsible for producing and regulating black bile.

All in all, the ideas around the four humors were contextual both to the practitioner and the time they lived in. Anything from diet treatments to experimental chemical influence could have been brought to bear.

EDIT: Here's a great listen on the subject http://www.maximumfun.org/sawbones/sawbones-four-humors

Follow-up: What was the the next course of action if the patient didn't improve with the standard remedy of their time?

Madness in the middle ages was not seen in the same type of medical light which we now perceive it. The way it was conceived of was representations of two separate forces, one as man in an animal state, in almost his state of nature which resulted from the Fall. Secondly it was perceived as a kind of other-worldly state of unreason, not unlike the phenomena of reason but corrupted by clinging to some kind of image of untruth. Foucault in Madness and Civilisation charts Western societies perception of madness and he argues that in the high middle ages and the renaissance, we can only understand their view of madness through these two lenses, that of a wild, unruly, unreasonable nature and that of the unknown and feared 'other' of Christianity be it death or the devil or other supernatural and feared imagery. He also argues that it wasn't until the Classical era that what we now call depression but what was then called melancholia emerged as a distinct category in the realm of madness.