Heisenberg uncertainty principle and lifetime of the excited state
For practical purposes the relevant aspect of the relation between the uncertainties in time and energy is that it is a Fourier relationship that explains how peak widths in a spectrum (with linewidths in frequency units) relate to the decay rates of time-dependent signals. A decay can be considered the statistical result of many independent events with an average probability of occurring described by a time constant $\tau$. It can further be shown via the Fourier transform that when a signal decays exponentially with time constant $\tau$, this leads to a Lorentzian peak shape with a width at half height proportional to $\tau^{-1}$. It is even simpler to show this in the case of a Gaussian shaped decay with linewidth ~$\sigma_t$. The resulting frequency peak is then Gaussian with linewidth ~$\sigma_t^{-1}$. When you multiply the linewidth parameters of the Gaussian functions you get: 

$$\sigma_f\sigma_t=1$$

If you use Planck's relation $E=h\nu$ if follows that

$$\sigma_E\sigma_t=h$$

Now - as mentioned earlier - you can think of the width of the Gaussian or Lorentzian time-dependent decay as proportional (or described by) a time constant $\tau$ (which is the default description in the case of exponential decays). It follows that 

$$\sigma_E~\tau \approx  h$$