For most of organized human society, there weren't censuses every 10 years or so. How do people guesstimate how many people were living in a place? I'm interested in the answer especially for places that didn't leave too many written records (I'm also guessing the answer would have something to do with midden pits or something like that?)
I can only answer this question specifically for Classical Greece.
The Greeks did not leave reliable census records, but the city-states did usually keep track of the number of adult male citizens eligible for military service. The paper strength of armies therefore provides the most reliable guide. In order to estimate the size of the whole population, the army figure given by our sources has to be multiplied based on how many men out of the total number would have been of military age (for this we tend to use standard population tables that are based on premodern European census records). This gives us the total number of male citizens including children and old men. That number then has to be roughly doubled to find the total number of citizens, male and female. To this we must add the number of resident foreigners and slaves, which are usually no more than estimates.
Needless to say, this is a pretty inexact method. It is, however, the best we can do. For a lot of Classical Greek communities we cannot do even this, because we're never told the paper strength of their armies.
Additional evidence can come from strange places. The size of fleets can be a big help; we know that the standard Greek warship, the trireme, had a crew of about 200, so that a fleet of, say, 60 ships would need a crew of 12,000. This can tell us a fair bit about the size of the population manning it. However, a lot of these crews may have been slaves or mercenaries, so it's difficult to be sure.
For Athens at the end of the 4th century BC, there are some inscriptions listing the people who enrolled in the state-funded military training programme. This tells us exactly how many male citizens were 18 years old in a given part of Athenian territory in a given year. From that we can extrapolate the total population using the comparative population tables mentioned above.
The final, least reliable and most controversial method is to estimate how many people a given area of farmland could feed, using the farming technology of the time. This is really hard to do properly because we don't know if all of it was used, how intensively it was used, whether food was being imported or exported, etc.
So yeah. Army sizes. Army sizes are the main demographic evidence for Ancient Greece.
That really depends on the time, place and available sources. /u/Iphikrates already mentioned a few methods. It's basically just a patchwork of the information available to try and come to a more accurate estimate. The earliest period I've done research on is probably 14th and 15th Century Flanders. There are a lot of trade and guild records available, so one popular method is to use these records in combination with other sources to try and accuratly guess the population in certain regions.
It's basically whatever you can get your hands on, depending on the region and time. I mostly know the correct terminology in other languages, but to name a few : Death records, food records, trade records, guild records, criminal records, literary sources, army inscriptions, inheritance records, estimates based on "news" sources, geographical information, topographical information, estimates based on assumed food production given the contemporary tools, etc.