Why do Orthodox Priests, Monks, and Bishops wear hats?

[Maria]

Why do Orthodox Priests, Monks, and Bishops wear those black hats with or without veils during certain parts of the Holy Services in Orthodoxy?

What are the names of these headcoverings, the history of these hats, and their function?

When they are worn?

Why are their different colors of these ecclesiastical hats? For example, I have seen black and purple hats.

In addition, regarding the Orthodox Bishop's Mitre, are there mitered priests among the True Orthodox?
Is this just an honor or does it bestow certain responsibilities?

[jgress]

Mitered priests only among the Russians, afaik. I don't know the answers to your other questions, sorry!

[Matthew]

To attempt to answer a couple of your questions but not all:

The skufos with veils are only worn by monastics.

The flat top hats I heard were made that way because the muslims on horses in Greece and other occupied Orthodox countries by the muslim turks used to whack them on the head as they passed by to squash them. So they decided to make them flat on purpose so that they would not be damaged later by muslims even if they were whacked.

I have noticed that clergy and monastics, when extra reverence is required the headdresses are taken off such as when anything holy like an icon, cross, relics are kissed, or any deep bowing or kneeling or act of humility and great reverence is done.

[jgress]

Also during the gospel reading. I'm sure there's a source somewhere that explains all this.

I'm not sure I buy the explanation for the flat cap. OrthodoxWiki is not very informative on the history, but suggests that it began in Byzantine times.

The Armenians wear pointed caps, and I imagine that Muslims would be as likely to whack them as Greeks!

[Matthew]

Yes, it may be a folk tale explanation. But who knows? It makes for a somewhat humourous tale, in any case!

[jgress]

Maybe all the whacking flattened them permanently.

[Maria]

Maybe all the whacking flattened them permanently.

And maybe, just maybe, those who had been whacked once did not get a second whacking.
So flattening all the hats prevented more whacking.