Finding how much time it takes for a complete Earth revolution around the Sun Long story short, my brother made a joke about how stupid it is to celebrate the Earth making one "trip" around the Sun: New Year's Eve.
So I got curious and was wondering: how could the first physicists guess that a revolution of the Earth around the Sun takes 365.25xxx days?
I've tried my luck on the internet but didn't find anything simple enough to understand. I've read about Kepler's law, and there are nice formulas, but how does he work this out? And how could the first people who knew that Earth was revolving around the Sun guess how much time an entire revolution took?
I asked my brother and he told me if he'd have to do it, he would simply watch the stars and map them, do that every night, and count the days until he came back to the exact same position of the first night. Is that realistic?
 A: 
I asked my friend, and he told me, if he'd have to do it, he would
  simply watch at the stars, and map them, and do that until he came
  back to the exact same position of the first night. Is that realistic
  ? I don't think the position would differ so much that the eye could
  see it. Am I right ?

No light pollution
Please remember there was no light pollution at that time. It was possible to see more stars than today, they appeared brighter and more colorful. The Milky Way, Andromeda Galaxy and Orion Nebula were perfectly visible to the naked eye.
Seasonal constellations
There are dozens of easy indicators in the night sky if you want to know the season:


*

*Arcturus, Spica, and Regulus form the Spring Triangle.

*Altair, Deneb and Vega form the Summer triangle

*Rigel, Aldebaran, Capella, Pollux, Procyon, and Sirius form the Winter Hexagon.


With this information alone, it is possible to reliably tell the current month by simply looking at the night sky after sunset, without any instrument.
Accuracy could be improved by using a very bright star (e.g. Sirius) and clear reference points (e.g. a far-away mountain).
Guardians of the pole star
You could use Ursa Minor and Ursa Major as an annual clock, with the two guardians rotating around Polaris. They do a full turn in a year, so an accuracy of 1° will give you an accuracy of ~1 day. It shouldn't be hard to measure with an astrolabe.

Improving accuracy
You cannot expect high accuracy if you measure the thickness of a single sheet of paper with a ruler. You can improve it greatly if you stack 1000 sheets, measure the total height and divide it by 1000.
Patterns in star position will repeat after approximately 365 days, but they will repeat almost exactly after 4 revolutions in 1461 days, which gives $\frac{1461}{4} = 365.25$.
 With enough time and dedication, it's possible to improve the accuracy even further.
A: In ancient civilizations, astronomy was a serious business (among other reasons, to accurately predict the seasons), so there were a lot of scientists making very careful measurements. Even with the naked eye, you can make quite accurate observations, and the ancients used these observations well.
The first really accurate determination of the length of the year was made by Hipparchus, a Greek astronomer who lived circa 190-120 BC. He calculated a year to be 365 + 1/4 - 1/300 (365.24667) days long, which is 6 minutes/year longer than the current estimate. Before that time, most people used 365.25, which is also not too far off (11 minutes), but is likely also due to the lucky coincidence the actual value is so near the neat round 1/4 day.
Hipparchus mostly used exact measuring of the equinoxes (the moment day and night are just as long, and when the Sun rises exactly in the east and sets exactly in the west). It's possible to measure this quite exactly, up to an hour. But Hipparchus also used the data gathered by earlier astronomers, back until 432 BC (~300 years before him). When you have this large time-span, and combine multiple measurements to cancel out any errors, it's thus possible to get very accurate results. Oh, and he used solar eclipses to refine his measurements as well, though the equinoxes were more important (there are more of them).
Incidentally, he was also the first one to discredit your question ;-)
That is, what most people think of as "a year", is actually not the time it takes the Earth to complete one orbit around the Sun. It's very close, but the Earth also precesses: it's top (the poles) very slightly turn around. It also means that Polaris (our current "northern pole star") will not remain the northern pole star forever (but it will for several thousand years). This also means that the time from equinox to equinox is not exactly the same as the time it takes the Earth to complete an orbit, it is shorter by 20 minutes and 24.5 seconds.
The time it takes the Earth to complete an orbit is called a sidereal year, but the time it takes for the equinoxes to repeat (and thus the seasons), is called a tropical year. Strictly speaking, "a year" could refer to both of them, but most of the time a tropical year is meant. And a tropical year is also what our calendar is based on.
At least, the western calendar uses a value of 365.2425 days, while the modern established value for a tropical year is 365.24219 days (27 seconds shorter), and a sidereal year is 365.256 36 days (1197.5 seconds longer)
These values are all assuming a day is exactly 86400 SI seconds, without any leap seconds.
A: Observe where the Sun rises and sets, and how high it gets in the sky at midday. These things repeat after 365 days.
A: It is not that simple. It takes tools such as Stonehenge to accurately time the passage of stars. You may then notice that the Sun's passage across the sky varies with a period of 365.25 days.
A: In order to find out how the topic evolved in the past 1000's of year, the proper search on the Internet is about the different calendar systems that people employed in the past. 
Generally, calendars evolved between better and better precision and different attempts to fit the seasons and the moon phases around some easy calculatable numbers (12, 30, 360, etc...).
It was late in the process when people found out about the movement of the celestial bodies.
A: If you would capture sun position in the sky at the same time each day of year - you would get a periodic figure, called Analemma which shows sun's path in a sky through a whole year :

A: It is a lot simpler. In Europe we have four seasons. The same season returns about 365 days later. Over time the measurement got more and more exact.
The reason we have seasons is that the earth axis is slightly inclined compared to the movement around the sun.
