There are three points: i) the exponential amplification in the PCR, ii) the generation of a (fluorescence) signal that is proportional to the amount of amplified molecules, and iii) the measure "ct" that is taken as the cycle number needed for the signal crossing a given threshold.

The exponential amplification of N0 starting molecules with efficiency E is given by

N(c) = N0*E^c

where c is the number of PCR cycles. The signal is proportional to this amount. Using p as proportionality factor the signal F at cycle c is

F(c) = p*N(c) = p*N0*E^c

The ct value is the cycle for which F = Ft (some arbitrary but assay-specific value):

Ft = F(ct) = p*N0*E^ct

This is it. Now you can ask yourself what a given ct-value tells you about N0:

Ft = p*N0*E^ct    

logE(Ft) = logE(p*N0*E^ct) = logE(p)+logE(N0)+ct  

logE(N0) = logE(Ft)-logE(p)-ct

The term logE(Ft)-logE(p) is an assay-specific constant.Let us denote this by u, so we get

logE(N0) = u-ct

Conclusion: the logarithm of N0 is proportional to -ct ("minus ct").

Now consider two genes A and B with initial concentrations A0 and B0 and measured with the assay-specific constants uA and uB. The ct-values are ctA and ctB, respectively. We are interested in the relative concentration A0/B0 (i.e. the concentration of gene A normalized to the conc. of B). We know:

logE(A0) = uA-ctA

logE(B0) = uB-ctB

If the amplificantion efficiencies for A and B are identical we can write

logE(A0) - logE(B0) = logE(A0/B0) = (uA-ctA) - (uB-ctB) =(ctB-ctA) - (uB-uA)

Conclusion: the log of the relative expression is proportional to the delta-ct value.

If you compare the relative expression between two contitions (trt, ctl), the fold-change (FC) of the normalized concentrations is

FC = ( A0[trt]/B0[trt] )  / ( A0[ctl]/B0[ctl] )

And again,

logEFC = logE( ( A0[trt]/B0[trt] ) / ( A0[ctl]/B0[ctl] )  )

            = logE(A0[trt]/B0[trt]) - logE(A0[ctl]/B0[ctl])

            = ((ctB-ctA)[trt] - (uB-uA)) - ((ctB-ctA)[ctl] - (uB-uA))

Now the assay-specific constants cancel out it simplifies to

logEFC = (ctB-ctA)[trt] - (ctB-ctA)[ctl]

Conclusion: the difference of the delta-ct values is the logarithm of the fold-change. The base of the logarithm is the amplification efficiency E.

There are three points: i) the exponential amplification in the PCR, ii) the generation of a (fluorescence) signal that is proportional to the amount of amplified molecules, and iii) the measure "ct" that is taken as the cycle number needed for the signal crossing a given threshold.

The exponential amplification of N0 starting molecules with efficiency E is given by

N(c) = N0*E^c

where c is the number of PCR cycles. The signal is proportional to this amount. Using p as proportionality factor the signal F at cycle c is

F(c) = p*N(c) = p*N0*E^c

The ct value is the cycle for which F = Ft (some arbitrary but assay-specific value):

Ft = F(ct) = p*N0*E^ct

This is it. Now you can ask yourself what a given ct-value tells you about N0:

Ft = p*N0*E^ct    

logE(Ft) = logE(p*N0*E^ct) = logE(p)+logE(N0)+ct  

logE(N0) = logE(Ft)-logE(p)-ct

The term logE(Ft)-logE(p) is an assay-specific constant.Let us denote this by u, so we get

logE(N0) = u-ct

Conclusion: the logarithm of N0 is proportional to -ct ("minus ct").

Now consider two genes A and B with initial concentrations A0 and B0 and measured with the assay-specific constants uA and uB. The ct-values are ctA and ctB, respectively. We are interested in the relative concentration A0/B0 (i.e. the concentration of gene A normalized to the conc. of B). We know:

logE(A0) = uA-ctA

logE(B0) = uB-ctB

If the amplificantion efficiencies for A and B are identical we can write

logE(A0) - logE(B0) = logE(A0/B0) = (uA-ctA) - (uB-ctB) =(ctB-ctA) - (uB-uA)

Conclusion: the log of the relative expression is proportional to the delta-ct value.

If you compare the relative expression between two contitions (trt, ctl), the fold-change (FC) of the normalized concentrations is

FC = ( A0[trt]/B0[trt] )  / ( A0[ctl]/B0[ctl] )

And again,

logEFC = logE( ( A0[trt]/B0[trt] ) / ( A0[ctl]/B0[ctl] )  )

            = logE(A0[trt]/B0[trt]) - logE(A0[ctl]/B0[ctl])

            = ((ctB-ctA)[trt] - (uB-uA)) - ((ctB-ctA)[ctl] - (uB-uA))

Now the assay-specific constants cancel out it simplifies to

logEFC = (ctB-ctA)[trt] - (ctB-ctA)[ctl]

Conclusion: the difference of the delta-ct values is the logarithm of the fold-change. The base of the logarithm is the amplification efficiency E.

Looks like classical music from a true master ;]

See the attached for several different ways of looking at this...

In addition (in simple terms):  slope of the standard curve = -1/log10(EAMP)

and EAMP = 10-1/slope and "Efficiency" (in terms of percent) = (10-1/slope) -1

Fold difference = EAMP(target)(Cq control - Cq treated) / EAMP(ref gene)(Cq control - Cq treated)

(Again - this is all in the most simplistic of terms).

And:  logEamp(serial dilution factor) = expected difference between Cq values of the dilution curve.  E.g., if the dilution curve (standard curve) demonstrates an EAMP of 1.8

(or 80% Efficiency), the slope is thus -1/log10(EAMP) = -1/log10(1.8) = -3.91738232676218, and, if you are using a serial dilution factor of 4, you would expect an average distance between Cq values on the curve to thus be: log1.8(4) = ~2.3585.  These relationships can be practiced ad nauseum in Excel until one gets a feel for them.  But, in reality, what Jochen has written above is the most responsible way to look at these relationships.