Like the Native Americans, the Egyptians placed great importance on the sun. Their main god, Amon-Ra, was said to hold the power of the sun. However, the calendar that the Egyptians used orginally was not a solar calendar. It was lunar, based on the phases of the moon. This calendar didn't always fit with the seasons, however, so every now and then, an extra month was added. How did they know when to add an extra month? They watched for the heliacal rising of Sirius.
Stars rise and set just like the sun does. The stars that are from our point of view will rise after the sun, so we can't see them. However, because the Earth revolves around the sun, the sun appears to move to the east through the stars. Therefore, eventually the stars that appear to be to the east of the sun will appear to be to the west of the sun, rising just before it does and being visible until sunrise. This is called heliacal rising. When the star Sirius rose in this manner during the last eleven days of the Egyptian year, they knew to add an extra month. The heliacal rising of Sirius also marked the time when the Nile river would flood and irrigate the farmland that the Egyptian society depended on.
The rising of all sorts of stars and constellations became very useful when the Egyptians switched to a solar calendar. Their solar months consisted of thirty days. Since thirty can't be evenly divided by four, they opted to have three ten-day "weeks". The beginning of these "weeks" was marked by the rising of certain stars or groups of stars called decans at nightfall. These decans were also used as a method of time keeping called a star clock. A star clock is simply a chart which had the name of each decan that rose on a certain date at a certain time of night. If you knew the date and saw that one of those decans was rising, you could use the star clock to find the time.
These decans also seemed to have a connection to Egyptian funeral rituals. Once a decan is no longer visible in the night sky, it does not rise heliacally again for seventy days. The Egyptians believed that during this time, the decan had died and was being purified in the Duat, or the underworld. After seventy days in the Duat, the decan was reborn. Apparently in correlation with that, Egyptian embalmers took seventy days to prepare a body for burial.
In most cases, the first thing one thinks of when one thinks Egypt is pyramids. The orientation of Egyptian pyramids suggests that their knowledge of geography and astronomy was pretty impressive. The four corners of most pyramids line up almost exactly with the four cardinal directions, only deviating by about six hundredths of a percent. It also seems that many pyramids are oriented to the three stars that make up Orion's belt. A theory known as the Star Correlation Theory points out that the "air shafts" are perfectly aligned to point to those stars, and to the star Sirius. Because the Earth's axis precesses (i.e. it makes a circle in the sky every twenty-six thousand years due to the gravity of the sun and moon), the stars don't appear in quite the same places as they did when the pyramids were built. There is a mathematical way of showing where the did appear, but it's a bit long to put in here. If you'd like to take a look at it, you can follow this link. (Well, you used to be able to follow that link, anyway. Unfortunately, the link is long since broken. Sorry!)