While Mark is trying to bring back images that remind you of the days when film ruled and film grain was a real consideration, there are many of us who try to remove excess grain. It is actually called "digital noise" and usually happens when you push your ISO beyond the sensor's optimal range. It can look like this:
See the random dots of color? Look at the band of his hat. This is from a D-80 at ISO 1250 - that is definitely beyond its optimal range. You'll always see the worst noise in the shadows, so this dark concert photo has major problems.
There are many programs and Photoshop plug-ins designed to reduce or eliminate this problem. I've used a couple of them, and they can make a significant improvement. Since I do most of my initial work in Lightroom (sorry to beat that drum, again), I use the noise reduction resident there. It can usually do just as good a job as the other programs.
You can find the sliders in the DEVELOP module, near the bottom right. I usually adjust the color first, then the luminance, because the color noise bothers me the most. Move the sliders slowly to watch for the improvement in image quality. The adjustment can only work so much magic. When you see the no further improvement, stop moving the slider. It does you no good to keep cranking on it. These two photos are processed identically, with the only difference being Lightroom noise reduction. You can see the image is still less than optimal, but there is a big improvement.
Some people really like the noise effects on a photo. It's a personal choice for you to make. I'm not saying you should never push the ISO higher. After all, it may be more important to get the photograph than worry about a little digital noise.
This last image was shot with the same camera at ISO 1000 - almost as high as the last image. The conditions weren't ideal - clouds and drizzle. It is brighter than a small stage in a bar (which means the shadows aren't as bad). I knew the darkness and the dark river were going to be problematic. My choice was to put the camera away or shoot with a high ISO. I chose to have a memory of a great trip to the Nantahala River. The noise is barely noticable.
So, I prefer no "grain" in most of my images, unless I've made the choice to have it there. You should always, however, push that ISO up when you need more to get the image. I use my tools to reduce noise most of the time, but don't let the worry of a little noise stop you from capturing the fun.