I am using Nikon D5000. Struggle to achieve correct exposure in low light or night portraiture. I tried to adjust exposure compensation manually in aperture mode,but could not get good result. Can anyone suggest an ideal mode for night or low light portraiture, and suggest ways to compensate exposure better.
There are two main concepts in play here: first, determining the "correct" exposure, and second, adjusting camera parameters to obtain that exposure.
The camera includes a light meter, which reads the scene and determines what is basically a programmed guess at "correct". It's important to realize that the camera has no way of distinguishing between a dark thing in bright lighting and a bright thing in dark lighting. (Some "matrix" metering modes try to guess at the situation, but it's really not all that smart. See When best to use Multi-Zone/Matrix, Spot, or Center-Weight? for a discussion of different metering modes.) That means that it basically just tries to make the subject a middle tone, no matter what.
Adjusting for different brightness of subject is what exposure compensation does: it's a way for you to tell the camera that its meter reading should be biased darker or lighter. If, for example, the scene is really dark, and you want a correspondingly dark photograph, you could use negative exposure compensation. See What is exposure compensation? and When should I use exposure compensation? for an in-depth discussion of this.
For the second part — camera parameters — you have three factors which you can adjust. These are lens aperture, shutter speed, and sensor ISO. For any given "exposure value" chosen by the metering, there are various combinations of these which will match. You could use a narrow aperture and long exposure, or a wider aperture and shorter exposure. Those choices will have different consequences for the resulting image, but the brightness will be the same. What you choose will depend significantly on the result you want. (For more on this, see What is the "exposure triangle"?, after the part where I rant a bit about the "triangle" terminology.)
For static subjects, a tripod and a long exposure is often the best approach, but if your subject is moving, you'll need a combination of wide aperture and high ISO. This may, in turn, result in blur and noise, but if it's really dark, there's no way around that. The only other solution is to actually add light — and if you're really doing "night portraiture" (as opposed to snapshots; not that there's anything wrong with that) that's probably what you actually want to do. (Take a look at How to light spontaneous portraits? for some suggestions on working with light when all you have is what's at hand.)
As you note that you're using aperture-priority mode, you might be interested in What is the relation between shutter speed and exposure compensation in aperture priority? to read more about what the camera does to the other exposure factors (shutter and ISO) when you adjust the exposure compensation. (Although the answer is simple: it changes them to obtain the total exposure value that matches the meter combined with your compensation amount, with the aperture fixed at what you've given.)
Overall, there are no magic settings here, and your Nikon DSLR is no different from any other camera in this regard. Photographs are literally made from light, so low-light photography is inherently hard and there's no getting around it.