This is the word that gave its initial letter, π, in Mathematics, from the meaning "circumference".
Also presumably the root of 'periphery' in English, which will help me remember it!
Would "suburb" be a possible translation, effectively a district on the periphery?
I don't think so as a suburb is a more specific type of district. The word for it follows the same pattern as in English: προ (before) +άστυ (=city) > προάστιο.
My problem with all these translations is that they don't include the meaning of the outskirts that I think is part of the word. I mean, could I use the meaning of περιφερεια to refer to the central district in Athens? If not then district is not a good translation and we need to think of some other word. If yes, then I understand this word.
All of Athens is in the περιφέρεια of Attica (together with some other places as well).
There are a total of 13 such περιφέρειες in Greece -- the Wikipedia article https://en.wikipedia.org/wiki/Administrative_regions_of_Greece calls them "administrative regions".
Translating the names of administrative divisions of countries is notoriously fraught since so many countries divide themselves up differently, even between countries that use the same language (the US has "states", for example, but the UK does not; on the other hand, the UK has "constituent countries" which the US does not).
So there's often no "obvious" translation for any particular administrative division, just tradition -- for example, calling the former divisions of Greece and those of current Japan "prefectures".