Lighting your subject
The flash built into compact digital cameras is only meant to throw light on a subject that's just a few feet away. In a lot of shots (especially when you're trying to get far enough away to get everyone in) the subjects can be much farther than the flash can reach. Pictures without enough flash tend to look washed-out and too dark, and many important details are lost.
Adding other sources of light will not only make your picture brighter but more interesting as well. And what better time than the holidays to find lots of alternate light sources! Try posing family members near a tree decorated with bright holiday lights, or snap a picture while they're lighting candles. The glow from a set of candles or string of lights can cast a nice, warm effect, as the light and shadows play across the faces of your subjects.
Get a little closer
Think about some of your favorite photographs of people. What makes them so appealing? Often the best photographs of people really say something about the subject's personality. Many portrait photographers prefer to take pictures of people they know, because they can use the familiarity with the subject to take a great picture. Many people get nervous with a camera around, but having a picture taken by a friend or family member can really help put them at ease.
A nice portrait during the holidays makes a great memento, which can also make a wonderful present. To take a great portrait shot, take a picture of a family member or friend doing something they love doing. If they like to cook, take a picture of them making the Thanksgiving meal. If they enjoy getting into costume, take a shot of them at Halloween.
Don't be afraid to get close to your subject and take lots of pictures. A good picture of someone in action makes for a wonderful portrait, and the more pictures you take, the better your chances of getting exactly the image you want.