The Wheel of the Year can mean different things to different Pagans - and to be sure, not all Pagans call their holidays by this name / celebrate the same holidays, or celebrate them in the same way. For many the Wheel refers to a series of eight holidays - the four quarters (the solstices and equinoxes), and the four cross-quarters (the points between solstice and equinox). It can also include the lunar cycles, as well. This series of eight holidays was likely not known as a whole to any ancient pagan group, and is instead a mash-up of a few ancient European calendars.
Before we get into the details, let's look a bit closer at some of these terms. The Wheel part is a fairly easy one to understand, as the year can be seen as cylindrical. Eventually we always come back to the same spot. It's October 18th today, a year from now it will be October 18th again, and a year from that we'll still be at October 18th. Since we're never truly at the same point in time again, some prefer to see the passage of time as a spiral. We started in the center, and each year we move out a ring. We come to the same point on the ring over and over, but it's always a new ring. The rings of a tree are also a good metaphor to examine.
What about "quarter" and "cross-quarter" though? Draw, or imagine, a circle. Then put a line down the center of the circle, vertically. Next, horizontally. The north/upper spoke on the wheel is the Winter Solstice. The southern/lower spoke it's counterpart, the Summer Solstice. The east/right point the Spring Equinox, and the west/left point the Autumn Equinox. Those are the four quarter holidays, and as you can see, the divide the year into four quarters. These holidays are usually called Yule, Ostara, Litha, and Mabon. (But remember, names can vary quite a bit!) Quarter Days are sometimes referred to as the "lesser Sabbats."
The solstices and equinoxes are not on fixed dates. Although they usually happen around the same times each year, the exact date can vary. The Summer Solstice was on the 21st of June this year, but will be the 20th of June next year.
Next come the cross-quarter days, these dates cross the quarters, so draw an X through the circle which splits it into 8 equal pie sections. These are the Holidays Imbolc, Beltane, Lammas, and Samhain. (Google image search "wheel of the year" for tons of pictures of this sort of chart.)
Since Samhain is the next approaching, let's use that for the next example. There are two ways to calculate the date for Samhain, and the other cross quarter days (sometimes called the "greater sabbats" or "fire festivals."). One is to go by fixed dates - Samhain would fall on the same date every year, usually starting at sundown on Oct 31st and running to sundown of Nov 1st. Why sundown? The Celtic calender is one the modern Wheel is based off of, and they began their day at sundown. (Similar to the Jewish and Islamic calenders.)
The other way to do this is to calculate the exact point between the Fall Equinox and Winter Solstice - this year that would be Nov 7th. Since the solstices and equinoxes shift slightly each year, so would these dates. Which method you use is a personal choice.
Now if you're in the Southern Hemisphere? Same thing, but flipped. While in the Northern Hemisphere we just celebrated the Autumn Equinox, in the Southern the Spring Equinox has just passed by - meaning they are getting ready to celebrate Beltane next, while in the North we are waiting for Samhain.