Paige-
When we specify mixed models we have fixed effects and random effects. The general equation for the model is Y=X*beta + Z*gamma + error. The betas are the fixed parameters and the random parameters. Now Z has to be a subset of X, namely the columns of Z are a subset of the columns of X. Also since the gamma are random they are distributed as N(0,G) where G is the covariance matrix of the gammas.
The interpretation of the beta parameters are regarded as population parameters and the gamma parameters are regarded as subject specific parameters. So for example lets say we were analyzing the effect of two treatments, a and b, and we randomly choose 10 clinics to select subjects for the study. Since the clinics are a random sample of all the clinics in the population we can regard the clinics as random effects. More to the point a simple model would look like:
Y=beta0+beta1*time+beta2*trt+gamma0_i + gamma1_i*time + error
So now each ith clinic adds their own intercept and time parameter to the model.
The conditional model is E(Y|gamma_i)=beta0+beta1*time+beta_2*trt+gamma0_i + gamma1_i*time. This model takes into account each clinic's involvement hence we are conditioning the Y given the gamma_i.
On the other hand the marginal model is E(Y)=beta0+beta1*time+beta2*trt. The random effect go away since in the marginal model we are averaging over all random effects and as was stated earlier the random effects have mean zero.
The residuals are the same as always. The predicted value Y - actual Y. But depending of whether you use the conditional or marginal model to predict Y we will be getting different values of the residuals.
Hopefully this helps. Let me know if I can clear anything up