Taking a decent portrait is extremely difficult with a point and shoot camera due to the limitations of the camera's aperture. Not only do DSLR cameras have a better aperture range than the typical point-and-shoot cameras, DSLRs support the use of multiple types of lenses (e.g., zoom, wide angle, low aperture).
For an indoor portrait I like using a lens with a large aperture. In the most basic terms aperture is the size of the hole in the lens that allows light to hit the lens sensor. Controlling the aperture is one of the three primary settings a photographer uses (the others being shutter speed and ISO). Depending on the conditions you will want to change the size of the aperture to change the amount of light entering the camera, which is especially important in low light settings.
As with any setting on a camera, changes come at an effect. In this case changing the aperture also changes the depth of field. Depth of field is the degree to which objects outside the primary focal point appear to be in focus. A shallow depth of field means that anything outside the plane of the primary focal point will appear out of focus, while a deep depth of field means that objects outside the primary focal plane will appear to be in focus.
So what does all of this have to do with taking a portrait indoors?
When taking portraits indoors, I like to open the aperture. The shallow depth of field creates greater emphasis on the subject of the photo by creating a slight haze in the background of the photo. The viewer understands instinctively the subject of the photo and is not distracted by the background. The background is visible and in many cases distinguishable, but not distracting.
Using a larger aperture has the side benefit of allowing for a slightly higher shutter speed and/or lower ISO or can negate the need for the use of a flash in lower light conditions. When taking a casual portrait indoors with a single subject I try to avoid using flash so as not to create additional shadows especially around the hairline. If I am using an external flash, I'll pointing the flash upwards (effectivley bouncing the light off the ceiling) so as not to create shadows around the subject and also lessen the chance of redeye
When setting up a portrait shoot I always try to bring the person away from reflective glass since these reflections can complicate the light scheme. I also try to make sure the subject isn't too close to a wall or background feature so that I can take full advantage of a shallow depth of field (the farther backgrounds are from the focal plain the more out of focus and hazed they will appear).
Most importantly, while focusing your shot I always use the subject's eyes as the focal point and then recompose the photo. As part of human nature, viewers are always directed to a subject's face, especially the eyes, so it's important to make sure that the facial features are well in focus.
Short tips:
Aperture: F2.8
Shutter speed: 60-80
ISO: >800
Flash: Angled high or Off
Focal points: One or few - on the eyes.