I am having a difficult time understanding what is large signal versus small signal analysis, the importance of either one of them. Also, how is DC gain of an amplifier different than the AC gain of a common emitter amplifier?
If take a small bit of the base-emitter characteristic, which is exponential, then you can say that this bit is a straight line and you can model the transistor with linear devices (current source and resistors-look for the pi model). The linear models of the transistors are valid only for small input signals because only a small portion of the exponential characteristic is used and thus is assumed to be linear.
Large signal analysis, means that you can't approximate the exponential characteristic with a straight line because you use a large portion of it. If you use the small signal transistor models the results will be inaccurate.
The DC gain is defined as HFE=IC/IB
IC=dc collector current
IB=dc base current
The AC gain is defined as hfe=ic/ib
ic=ac collector current
ib=ac base current
Since the DC and AC currents are different, then DC and AC gains are different. This is a simple explanation, what happens in reality is much complex.