Is it how long it takes the Earth to go around the Sun once or what the calendar says?
The former.
It depends… are you talking about a calendar year or an astronomical year?
A calendar year is 365 (or 366) days. No more and no less.
The earth, however, revolves around the sun every 365 1/4 days yeah, I know that’s not exact… That is an astronomical year.
Zev Steinhardt
It takes the earth 365 days, 6 hours, 9 minutes and 10 seconds to make a complete orbit around the sun, coming back to the exact point where it started, relative to the positions of distant stars. This is known as a sidereal year.
A tropical year, or the amount of time from vernal equinox to vernal equinox, is 365 days, 5 hours, 48 minutes, and 46 seconds. This is due to the wobble of the earth’s axis and the precession (shift) of the equinox.
Yes, there are many different “years” to an astronomer. In addition to the two that h.sapiens lists, there is the “anomalistic” year and the “eclipse” year. Don’t worry, though. It’s not as bad as it sounds. These specialized years are only used for specific purposes. It’s not like if you ask an astronomer how long a year is, she’ll get confused and ask for more information.
Zev: Depends on which calendar you’re using.
This site gives information on the various definitions of year:
Take your pick.
This question made me wonder. About the astronomical year, that is. Is it the galactic year or the universal year? Just as the sidereal day differs by about 4 minutes from the average solar day because the earth revolves around the sun and therefore the appears in a different position against the “fixed” stars at the end of each solar day, so the solar system revolves around the center of the galaxy (with a period of 50 million years, IIRC) and so faces the “fixed” galaxies from a slightly different vantage each year. I figure the difference is about 3/5 of a second a year, but that is the sort of correction that the astronomers make when they insert a leap second every year or two. So what is it? The teeming one wants to know.
It is an “Earth year”, not even a “solar year”, since it is simply measuring how long it takes the earth to circle the sun. A “Venus year” would be shorter and a “Pluto year” much longer, but as far as I know we would still measure them in earthly terms. 
The question was, how do you know it’s made one complete circle of the sun? When it’s come back to the same position (angle) relative to the Galaxy, or relative to the Universe? As Hari Seldon said there is a measureble difference between the two.
I’m not 100% certain but I believe it’s relative to the Universe. Quasars are often used as reference frames. They are farther away than anything else we can observe, and they are point sources so position measurement is not ambiguous.
It occurs to me that there is yet another year, the most important for biological activity, and that is the year from one winter solstice to the next. That year presumably differs from the others by some relatively insignificant, but still measurable, amount. That is the one whose drift caused the Gregorian calendar reform, which was caused by the difference between this solstitice year and 365.25 day year, which made planting dates wrong.
The year from solstice to solstice is presumably the same as the year from equinox to equinox (the tropical year). Or am I being too simplistic here?
Yes, UDS, you are. The time from one vernal equinox to the next is 365.242374 days. The time from one winter solstice to the next is 365.242740 days. The difference is caused by regression of the Earth’s perihelion. See “Marking Time” by Duncan Steel for a thorough discussion of the many different types of astronomical and calendrical years.