In ancient Rome, before a major decision, the augur selected an area of high elevation with as broad a view as possible. He drew a foursquare design aligned with the cardinal directions. This space and it sky above it were called the templum. The augur put up a square tent at the center of the space and called it the tabernaculum. The entrance to the tabernaculum faced south. The augur sat in the entrance looking southward. To the east, the area of light; to the west, the area of darkness.
The augur's job was to watch the birds for omens—messages from the gods. Signs from the east (the augur's left) were lucky and prosperous. Signs from the right side were unlucky. The practice of observing birds for omens was called auspicum, which gives us the word auspicious.
The Latin word dexter means skillful or right (as in right hand), and comes from the Proto-Indo-European root *deks- meaning on the right or south. Sinister is the Latin word for left (origin uncertain).
The early practice of augury came from ancient Greece, where the augur sat facing north instead of south. Omens from the left were therefore from the west—the darkness—and portended unfavorably. The Greek practice influenced the Roman practice, but true Roman augurs faced south. In ancient Rome, sinister meant both favorable and unfavorable at different times, depending on whom you might ask.
In Old English, the right hand was the swiþra (stronger) hand. In the 12th century the word riht (good, proper, straight) came to mean right (as in right hand) based on the idea that the right hand is the proper hand. The word left probably comes from northern Old English *lyft (weak).
Although we don't use the words dexter and sinister for our right and left hands, they live on as dexterity (skill) and sinister (malicious and misleading). Left-handers should remember that to a true Roman augur, the left hand is auspicious.
You'll get no augument from me!