The days of purification are placed on the 18th and 19th of every month of the Hellenic festival calendar, and are thus part of the Mên kata Theion, the sacred month--religious events that return every month in a set sequence. These days of purification are hinted at in Proklos' 'Commentary on Hesiod's Works and Days', and further solidified by Philokhoros in his 'On Days'. The text is not available online in translation, as far as I can find, but in short, these texts indicate that 'the eighteenth and nineteenth of the month are devoted to purificatory and apotropaic rites.'

While this can be interpreted as a monthly thing, in context it can also be interpreted as something belonging only in Boedromion, tied to the Mysteries at Eleusis. Because we have no conclusive evidence in favor of either theory, I prefer to keep these as a monthly event.

The impure days are the days before the Deipnon. One of the most important of the many Hellenic festivals is the three-day transition from month to month. Although unlinked, the Deipnon, the Noumenia and Agathós Daímōn are held on consecutive days, around the new moon. The Deipnon (Hene kai Nea)--or Hekate's Deipnon--is celebrated any time before the first sliver of the new moon is visible. In practice, this is the day after the new moon. The impure days are thus the two or three (depending on if the month is hollow or not) days leading up to the Deipnon. These days are labelled 'impure' as a precausting. The Deipnon itself is absolutely impure, after all.

Hekate’s Deipnon is the traditional time to end the old month and prepare for the new one. In ancient Hellas, the Deipnon was celebrated with a Supper for the Titan Hekate--made up of leek, egg, cakes, fish, unions and garlic--and set out at the outside shrine to Hekate, and then placed at a crossroads as an offering to Her--Hekate Trioditis, Goddess of Crossroads--and the vengeful spirits who were in her following. In addition to placation, the Deipnon is also a time of purification; in ancient times, a dog was taken in, touched by all members of the oikos so any lingering miasma was transfered to the dog. The dog was then sacrificed in a holókaustos. This was most likely not a monthly thing, but only performed when the household was troubled. As Hekate's sacred animal is a dog, the sacrifice also served to regain, or keep, Hekate's favor upon the household. The house was also thoroughly cleaned so the new month could start fresh the day after. Debts were repaid on this day.

Purificatory rites, obviously, refer to katharmos--ritual purification--in relation to miasma--ritual impurity. Within Hellenic practice, miasma (Μίασμα) describes the lingering aura of uncleanliness in regards to a person or space through which contact is made with the Gods. Miasma occurs whenever the space or person comes into contact with death, sickness, birth, sex, excessive negative emotions and bodily fluids. It also comes from a lack of contact with the Hellenic Gods. Not the actual acts of dying, sex and birth cause miasma but the opening up of the way to the Underworld (with births and deaths) as well as contact with sweat, blood, semen, menstrual blood and urine pollutes us. Miasma is an incredibly complicated and involved practice and it's often misunderstood. The most important things to remember about miasma is that it holds no judgment from the Gods, and that everyone attracts miasma. It's a mortal, human, thing.

So let's look at the terms of 'purification' and 'apotropaic' as noted in the ancient texts. The practice of purification is called katharmos (Καθαρμός). The process of katharmos is elaborate because the process not only involves the physical but also the emotional, mental and spiritual. It means being physically clean when entering ritual but, peraps more importantly, it means leaving behind negativity, worry, pain and trouble before getting in contact with the Gods.
Katharmos is a Hellenic basic. There can not be ritual without katharmos. As such, it makes sense to have days in the month assigned especially to getting clean before the rest of the month's rituals take place.

The word 'apotropaic' comes from Greek apotrepein 'to ward off' from apo- 'away' and trepein 'to turn'. These practices serve to appease polluting spirits, ghosts chief amongst them. Ghosts were the people who could not find the entrance to the Underworld or who didn't have the money to pay Kharon for their passage. Those who were not properly buried were also doomed to wander the Earth for a hundred years. Interestingly enough, Hellenic heroes were also considered ghosts and were honored in the same type of rites as other types of ghosts.

The ancient Hellens held festivals in honor of ghosts, and the Theoi that presided over them, so they would be sated and appeased and would not haunt them. Most of these festivals included a holókaustos--a sacrificial offering given in its entirety to the Gods--and were solemn affairs, conducted at night and without an offering of wine. It was, however, also possible to invite them to dine with the living so they would feel included and then end the proceedings by asking them to leave; something the ghosts would do as it's not polite by the rules of xenia to be a burden on your host.

This fear of spirits and other supernatural entities was named 'deisidaimonia' (δεισιδαιμονία). The ceremonies of riddance were known to the Hellenes as apopompai (ἀποποπμαί), 'sendings away'. There isn't a single word in the English language that conveys the practice. Closest would be 'exorcism'. It is important to note here that these 'exorcisms' weren't performed on people, but on the ghosts themselves and in short, they consisted of rites to ask the Gods (especially Hekate) to keep these unfortunate souls away from their homes and families, because the ghosts could bring misery down upon them. Possession was not part of the fear. Many rites in the ancient Hellenic religion--including monthly ones like the Deipnon--were apotropaic.

There is one other reson I can think of why certain days could have become impure are the ancient Hellenic homicide courts. Homicide courts were the law cases where one man stood accused of murdering another. In many respects, a homicide case resembled other private suits. Each litigant pleaded his own case in two speeches, the first of which presented the main points in his accusation or defense and might be written for him by a logographer, aprofessional speechwriter, and the second rebutting his opponents arguments. Witnesses could testify if they were male citizens. When the litigants finished their two speeches, the jury voted for acquittal or conviction and the majority carried the day (a tie vote meant acquittal). The penalty for intentional homicide was death, though exile seems to have been a common outcome, and the accused was allowed to go into exile voluntarily at any time up until his second speech in court, which would then be delivered by a friend or relative in the hope of persuading the jury to vote for acquittal despite the accused’s departure. For unintentional homicide the penalty was exile, probably for a specific length of time (perhaps a year). At any point the victim’s family could agree to a lesser penalty, or they could even drop the charges, if they wished, though there was a strong moral obligation to avenge an intentional murder.

Interestingly enough, homicide courts were held outdoors out of fear of pollution. The belief was archaic and had become more of a tradition than a rule by the time of Classical Greece, but it was definitely practiced. It could have been that these homicide courts were always planned on impure days--or that these days became impure because these very specific courts were always held on these days.

So, what do these days mean for worship...? Not too much. Festivals were hardly ever celebrated on these days and essembly meetings were rarely held on them either. They were said to bring bad luck; no major endeavor was started on them. Especially the impure days in the middle of the month were considered ominous. For us, we can foolow the lead of the ancient Hellenes: there are no planned festivals and we can be mindful of these days when planning our own celebrations. But the ancient Hellenes would have held regular household rituals on these days and in case of imergency, they would have performed rituals as well. The katharmic rites might have been a bit longer or performed with more intention but overall, these days are just something to keep in mind, not be held back back.