In photographic terms, noise is the digital equivalent of grain produced by a film negative.
Noise appears on the image as minute color aberrations at the pixel level that gives the impression of tiny dots across the image. The image below shows an extreme example of noise on a tiny part of a digital image.
Noise typically occurs when the exposure is too long, too shot, or in some instances at low or high temperatures. Use of a high ISO (greater than 800 - or on really good cameras, greater than 1600 ISO) is another factor in how much noise the camera generates.
Cameras with a larger sensor (known as Full Frame cameras) tend to produce less noise because the sensor has a larger area on which to collect the data for the picture.
Noise reduction software and features are available. I like the Nik software Dfine2 or even Adobe's Luminance setting in Lightroom. Using noise reduction usually results in a softer image.