In some cases (as in the microwave cavity resonators or vibrating strings) the system has many characteristic frequencies and can show resonance at many frequencies.
As a general fact, the external "force" supplies energy to the system producing an increase in the amplitude of the "oscillations". This energy input to the system is more efficiently produced when the external force acts in a synchronous way, and this is achieved when the external frequency is close to the natural free frequency of the system. Otherwise the system sometimes gains energy and sometimes gives it back to the external source.
The system is said to be at
resonance when both frequencies, the characteristic or natural of the system
and the external one, are coincident. If the system has no possibility
of releasing energy to the environment, that is to say, is a loss free
system, the oscillations would increase without limit and the system eventually
would break. But in general there are mechanisms for releasing energy to
the exterior (like friction in the mechanical systems, or resistors in
the electrical circuits). In this case, when the system is at resonance,
the largest amplitude can be observed, but in the stationary state, this
amplitude is stabilized when the energy input from the external source
is equal to the energy lost to the environment.