Grain in film photography is similar to noise in digital photography. Just like using film with higher sensitivity (or higher ISO) results in larger grain on your images, you will get more noise when you increase the ISO setting on your digital camera.
A general rule is to use the lowest ISO setting possible to avoid noise in your pictures. This is especially true for most point and shoot cameras because their smaller sensor is more prone to produce noise at higher ISO settings than DSLR cameras with larger sensors.
However, more noise can also add to the mood of a photo, especially with black & white photos.
You could simply increase the ISO setting. This will result in more noise in your images, but you will have no control over the appearance of the noise.
To have more control over the appearance of the noise, you can add the noise during post-processing:
The noise is now added to your photo and you can adjust the amount and appearance of it:
none
For more information on this topic, you can also listen to the following TFTTF shows:
* TFTTF show 19 - Grain or
* TFTTF show 89 - Why higher ISO can be better for some background information on the origin of noise in digital images