Having owned a real narrowboat until quite recently, I would agree with playmofire that, at least in a British context, the term applies to boats on the canal system, some of which uses rivers to link canals, confusingly. For example, if you want to go from Bristol to Birmingham, one route is to travel down the River Avon, up the River Severn, along the Gloucester-Sharpness Ship canal to Gloucester, onto the River Severn again, up to Tewkesbury, then up the River Avon (a different one) to rejoin the canal system near Stratford. Or from Tewesbury, you can carry on up the River Severn to Stourport and get onto the canal system there. The rivers are often too dangerous to take a flat-bottomed narrow boat on, e.g. during the winter floods.
Narrowboats are sometimes called canal barges and (erroneously, unless you are a Viking) longboats. Some canals can accommodate wide-beam boats which aren't very narrow at all. Just to add to the confusion!
