Fourier transform and impulse function $\delta(\omega)$
The Dirac delta impulse $\delta(\omega-\omega_0)$ represents a spectral line at frequency $\omega_0$, since it is zero everywhere except for $\omega=\omega_0$. So any function with spectral lines, such as a sinusoid, or a DC signal (which has a spectral line at frequency $\omega_0=0$) has a Fourier transform which contains Dirac delta impulses. This is also true for any function with a DC component such as the unit step $u(t)$. Note that also periodic functions have Fourier transforms consisting of Dirac impulses because they can be represented as a sum of sinusoids (Fourier series).