Logistic regression is used to model probabilities. Say you were interested in an outcome of a binary nature (0,1) whose probability of occurance was p. One way to model this probability is the linear probability model:

p = a + bx + e

in a similar manner to standard linear regression. Note that in this case, a + bx is not bounded (can go from -inf to inf), but clearly our probability must be greater than 0 and less than one. One way to fix this problem is to use the logit of p:

logit(p) = log(p/(1-p)) = a + bx

the logit transforms the probability onto the entire real line, so we dont have the problem with the linear probability model. We can solve for p in this function by:

p = exp(a+bx) / (1+exp(a+bx))

the plot p as this function of x takes on the 'S' shape that I believe you associate with a saturation effect.

Now, If you are not interested in the probability model, but only want to model the assocation between an outcome y and a covariate x whose relationship appears to be 'S' shaped, then any 'S' shaped function may be used for a model such as:

y = c + d / (1+exp(a+bx))

where a,b,c, and d are parameters.

~Matt