Shadows are the evil nemesis of every photographer, and during the age old battle, many devices have come to help fight the good fight. One of those is the Speedlite. By attaching it to the camera, the photographer can usually avoid a high contrast look (remember: high contrast = a lot of shadows. It may be good for Hitchcock, but not for you cousin's Bar Mitzvah). A speedlite, when used correctly, can be a fast and easy solution to the shadow problem and can save you the hassle of working harder in post production.
Essentially an external flash will allow you to "bounce" the light of the flash off other objects, like the ceiling or wall, and onto the subject of the photograph. The swivel technique is a good place to start. Pointing the speedlite directly at the subject will be most unflattering and will create harsh shadows, exactly what you're trying to avoid. Position yourself so there is a wall either to the left or right of the subject. Swivel the speedlite to the 3 o'clock position (if the wall is on your right) or 9 o'clock (if the wall is on your left). The speedlite will be aimed at the wall while your camera is aimed at the subject, creating a side lighting effect. If you swivel it to a wall directly behind you instead, this will create wrap-around light, covering the subject very evenly and eliminating almost all shadows.