Length of stay as tourist in another European country, as Schengen/EU/EEA citizen

The key limit is three months. It has nothing to do with Schengen but is part of general EU law. As such, it also applies outside the Schengen area (i.e. to the UK, Ireland, Romania, Bulgaria, Cyprus, Croatia – and to Switzerland through a distinct agreement that was implemented even before the country became part of the Schengen area).

Even in countries where there is no special formality to register yourself, stays longer than 3 months do not follow the same rules. You have a very strong right to travel for tourism and to stay for less than 3 months but no unconditional right to stay elsewhere in the EU for longer than 3 months (see this answer to another question for the legal basis for this distinction).

In most places, you don't need to do anything for short stays but even though it gives you the right to visit other countries, EU law does not strictly forbid requirements to report your presence, even if you don't intend to become a resident. Europa.eu provides practical information on this but even this site (based on information submitted by the member states themselves) is incomplete and at times vague so it's difficult to know exactly what the requirements are country by country.

Also note that residence is often defined by intent and by other factors like the location of the “center of your interests” (income, property, family…). So residency does not necessarily start after the first three months have elapsed nor does it depend solely on the length of stay.

For example, if you sell your house in France, move your stuff to Germany and rent a flat there, then you are required to register within a week or two of moving in your new home, even if you are careful to never spend more than three months in the country. On the other hand, if you have a house in France and stay at a hotel in Germany for two months, you will probably not be considered a resident.

If you would do the same the other way around (i.e. move to France), you would not be required to register at all (because that's not necessary in France in the first place) but you would still be deemed a resident from the date you actually entered the country (and could e.g. have to pay taxes based on that).