Does a "frequency-modulated" signal mean that the frequency isn't consistent? and therefore it has a specific bandwidth of frequencies is can cover??
Frequency modulated means the instantaneous frequency of the signal is varied, with the instantaneous frequency carrying the information in the signal. This instantaneous frequency will only be of a limited range. If you provide more context we could help more.
therefore it has a specific bandwidth of frequencies is can cover
This is true of all signals. All signals have a specific bandwidth (aka range of frequency components that make up the signal via fourier transforms). Even amplitude modulated signals where the instantaneous frequency is constant (the carrier) have a bandwidth of frequency components that compose the signal.
A frequency modulated signal means that the frequency is shifted around a carrier frequency. This can be done digitally or via analog multiplication.
What it looks like is a sine wave speeding up or slowing down.
Yes, it normally has a specific bandwidth, some of the reasons being is radio licenses and bandwidth are in short supply (there are a limited number of frequency ranges). With frequeny modulation one can transmit more information in a given section of bandwidth. Most digital wireless communication uses frequency modulation.