the definitions between narrowband and wideband amplifiers is dependent from the frequency and the bandwith you want amplify.
For Example:
You have an AF - Amplifier and you want to amplify the frequencies from 10Hz to 30kHz, you can call it wideband amp. If you want to amplify only the speechband from 300Hz to 3000Hz it's called "Narrowband amp", because you only amplified a small part of the whole AF -band.
Narrow band means you amplify only one or some frequencies . for example we've an amplifier which can amplify only 1 GHZ or only 1 HZ both are narrow bands . but if an amplifier can amplify more than some signals for example if an amplifier can amplify 20HZ to 20 KHZ or 1 MHZ to 200 MHZ ... etc both have been called wide band amplifiers .
In short form : narrow band in amplifiers means limited frequencies that an amplifier can amplify .
Narrow band : that amplifies low range of frequencies, no Standard of how much bandwidth can be termed as narrow band.
Wide Band: Amplifies wider bandwidth signal, difficult to design, owing to the fact that u have to keep gain and matching high and constant for a very big band, keeping both gain and bandwidth high at the same time reuires expertise and hell of knowledge.