How does Prism compute the standard errors (and confidence intervals) of the best-fit parameters in nonlinear regression? And how does it calculate the confidence interval of the ratio or difference of two parameters?

Prism computes these values in a standard way. These use a mathematical simplification, so are considered approximate or asympotic values.

Standard error of parameters
Each parameter's standard error is computed using this equation:

SE(Pi) = sqrt[ (SS/DF) * Cov(i,i) ] 

where:
  Pi : i-th adjustable(non-constant) parameter
  SS : sum of squared residuals
  DF : degrees of freedom (the number of data points minus number of parameters fit by regression)
  Cov(i,i) : i-th diagonal element of covariance matrix
  sqrt() : square root
 

Confidence intervals of parameters
The 95% confidence intervals are computed by this equation:

From [BestFit(Pi)- t(95%,DF)*SE(Pi)]  TO  [BestFit(Pi)+ t(95%,DF)*SE(Pi)] 

where:
  BestFit(Pi) is the best fit value for the i-th parameter
  t is the value from the t distribution for 95% confidence and DF degrees of freedom.
     Example with Excel for 95% confidence (so alpha = 0.05) and 23 degrees of freedom: 
            = TINV(0.05,23)
  DF equals degrees of freedom (the number of data points minus number of parameters fit by regression)

For a discussion, see chapters 16-20 in Fitting Models to Biological Data using Linear and Nonlinear Regression.

Confidence intervals of difference or ratio of two parameters

When you enter a user-defined equation, you can ask Prism 5 to report the difference between two parameters, or the ratio of two parameters, and it reports this with a 95% confidence interval. This document  explains the calculations.