Wikipedia defined Resonance as the following : Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. What is meant by : (or a Fourier component of it) ?
Wikipedia Link : https://en.wikipedia.org/wiki/Resonance
It means that the driving force is decomposed into sines and cosines and if one of this sinusoidal functions oscillates with frequency equal or close to the natural frequency, then the system resonates.
To see this, consider for example an oscillator described by $$\ddot x+2\gamma \dot x+\omega_0^2x=\frac{F(t)}{m},$$ when $F$ is an arbitrary periodic driving force, of period $T$. It can be written as a sum of sines and cosines through a Fourier decomposition, $$F(t)=\frac{a_0}{2}+\sum_{n=1}^\infty\left[ a_n\cos\left(\frac{2\pi nt}{T}\right)+b_n\sin\left(\frac{2\pi nt}{T}\right)\right].$$ Then the particular solution is $$x(t)=\frac{a_0}{2k}+\frac{F_0}{m}\sum_{n=1}^\infty\left[\frac{a_n\cos(2\pi nt/T)+b_n\sin(2\pi nt/T)}{\sqrt{(\omega_0^2-4\pi ^2n^2/T^2)^2+4\gamma^24\pi^2n^2/T^2}}\right],$$ and there is energy resonance when $2\pi n/T=\omega_0$ and amplitude resonance when $2\pi n/T=\sqrt{\omega_0^2-4\gamma^2}$. The amplitude of the $n$-mode of $x$ gets its maximum value or is close to it.
