Constant trend removal either adds or subtracts a constant value to the whole surface (if it is done globally). This value is the same everywhere in the surface. A first-order trend removal will use the X,Y coordinates as predictors in a first-order linear model.
Constant: z = Intercept + Error
First-order: z = Intercept + aX + bY + Error
...where z is the variable you are interpolating, and a,b are coefficients that are calculated using ordinary least-squares. Kriging is then performed on the Error term. When the trend removal is done globally (as in the Kriging tool in Spatial Analyst), the values of [Intercept, a, b] will be the same across the entire surface. When you do a local trend removal (only in Geostatistical Analyst), the values of [Intercept, a, b] will be calculated locally using weighted least-squares, so the coefficients will change across the surface.