The only SI unit of time is the second (s) , but many other units are commonly used. (See Note)
Note that we also use the word 'day' to mean the period between Sunrise and Sunset, for example 'the days are longer in the Summer than the Winter'. Usually it is quite clear which meaning is intended.
A calendar year must have a whole number of days in it so is slightly different from a solar year, which is the time taken for the Earth to go round the Sun.
A calendar month may have 28, 29, 30 or 31 days and so is not a very precise measurement of time. Calendar months are very seldom used by scientists. Lunar months formed the basis of the very earliest calendars, but are not used at all by modern scientists.
Originally the most accurate clocks were based upon
measurements of the movement of the Sun and stars, but now
they are based upon the vibrations of atoms or crystals. One
of the most accurate clocks in the world is the caesium
atomic clock at the National Physical Laboratory at Teddington in West
London. This is accurate to one second in a hundred million
years. This clock produces a signal which is sent to a
radio transmitter at Rugby, and you can buy clocks and watches
which are radio-controlled by this signal and so keep time to
the same accuracy. You can visit the NPL web site, which
gives a lot of information on time and related ideas, by
clicking here 
Many other modern watches and clocks use the vibrations of a quartz crystal to keep time - in 2002 you can buy a quartz watch for less than £5 which keeps time to within a second a year, whereas in 1958 you had to pay nearly £100 (at least £1000 at 2002 prices) for a watch which kept time to within ten seconds a week!
Most of the people who made the astronomical measurements upon which our modern calendar is based were not Europeans, but they did live between the Tropic of Cancer and the Arctic Circle, and it is their view of the motion of the Sun and Moon which is described in this article. Many other ancient peoples living in the tropics, polar regions and Southern hemisphere also made astronomical observations and developed calendars, but these did not contribute to our modern calendar.