Think of it as the frequency of free vibration. If you grab your car ratio antenna, move it, and then let it go, it will vibrate at a certain frequency and with a certain shape. This frequency and shape would be very close approximations to the natural frequency and natural mode shape.
A structure has many natural modes. Often, the one with the lowest natural frequency is the main one we care about, but not always. In my car antenna example, that frequency and shape would be just one of the natural modes.
If a structure is pushed at a certain frequency, it vibrates at that excitation frequency, not the natural frequency--need to keep that straight.