Enveloping is used when you only really care about the peaks of a signal. For example, consider an impact test where you observed the damped free vibration of an oscillator and say you want to estimate the damping. You might not care so much about all sinusoidal stuff, you just want to know how quickly the amplitude is dropping. One way to find an envelope is to locate the peaks of the signal and then fit a cubic spline between them. This works okay.
Perhaps a better way is to use a Hilbert transform. In matlab, the envelope of a signal is given by abs(hilbert(x)). While this is useful, I am somewhat displeased with Matlab's naming convention because the function hilbert() does not actually compute the Hilbert transform. imag(hilbert(x)) is the Hilbert tranform. What hilbert() actually returns is the analytic signal (not be confused with an analytic function in calculus). An analytic signal has the property that the negative frequency components of the Fourier transform are all zero (obviously, real-valued signals are not analytic). Matlab uses a DFT approximation to the Hilbert transform. See this page for more information [10]
