Todd,
Nonstationarity is a second-order spatial effect and is commonly referred to as "pseudo-autocorrelation". Often nonstationarity in ecological models is caused by a spatial dependency on an auxiliary variable. Moran's-I is a statistic used to identify first-order spatial effects. It is important to understand these distinctions before interpreting the results of a exploratory spatial data analysis. You can think of first-order effects as global and second-order as local. In traditional geostatistics there are a few models of nonstationarity. The most conservative is stationary mean and variance and the most relaxed is a stationary mean. You can explore nonstationarity using the local LISA or Getis-Ord (Gi*) statistics.
I do disagree with Lauren in that if your residuals appear heteroscedastic this is often indicative of autocorrelation or non-independence in your model but not always spatial dependency. It is a fallacy that residual structure indicate model misspecification. You can also have the variable that is causing a spatial dependency present in the regression and still observe structure in your residuals, as you would expect. The mere presence of variables exhibiting first or second order effects does not detrend the effect in an OLS. GWR is a regression model that is appropriate when you have a second order spatial effect and was designed specifically to account for nonstationarity. GWR does not "reduce spatial autocorrelation" but rather incorporates it. If autocorrelation is due to first-order effects then a spatial regression or spatial autoregressive model is called for (or even a OLS with a spatial lag term). Since GWR is a local regression, akin to ridge or kernel regression, it does a poor job at estimating first order trend. However, with second order spatial effects one would expect to have a fairly non-linear relationship, making GWR very attractive. Alternative methods would be conditional autoregressive model (CAR), non-parametric (Random Forests) or semi-parametric (Spline regression) models.
Spatial statistics is a rather large sub-discipline of statistics and a complex one at that. While ESRI has developed some very nice tools for exploratory analysis I would highly encourage you to read some of the primary literature and consider using a statistical package (e.g., R, S+) that is better suited for this type of analysis.
Some good places to start in understanding spatial statistics are:
Fortin, M-J & M. Dale (2006) Spatial Analysis: A Guide for Ecologist. Cambridge University Press.
Haining, R. (2003) Spatial Data Analysis: Theory and Practice. Cambridge University Press.
Isaaks, E.H. & R.M. Srivastava (1989) An Introduction to Applied Geostatistics. Oxford University Press.
