Meaning of Hilbert Transform
One application of the Hilbert Transform is to obtain a so-called Analytic Signal. For signal $s(t)$, its Hilbert Transform $\hat{s}(t)$ is defined as a composition:

$$s_A(t)=s(t)+j\hat{s}(t) $$

The Analytic Signal that we obtain is complex valued, therefore we can express it in exponential notation:

$$s_A(t)=A(t)e^{j\psi(t)}$$

where:

$A(t)$ is the instantaneous amplitude (envelope)

$\psi(t)$ is the instantaneous phase.



So how are these helpful?

The instantaneous amplitude can be useful in many cases (it is widely used for finding the envelope of simple harmonic signals). Here is an example for an impulse response:



Secondly, based on the phase, we can calculate the instantaneous frequency:

$$f(t)=\dfrac{1}{2\pi}\dfrac{d\psi}{dt}(t)$$

Which is again helpful in many applications, such as frequency detection of a sweeping tone, rotating engines, etc.



Other examples of usage include:


Sampling of narrowband signals in telecommunications (mostly using Hilbert filters).
Medical imaging.
Array processing for Direction of Arrival.
System response analysis.