If you want to test whether Z moderates the relationship between X(IV) and Y(outcome) create a new variable by multiplying Z*X. This is your interaction term. Enter X, Z, and X*Z as predictors into a regression with Y as the dependent variable. The coefficient for X*Z tells you how much the coefficient for the relationship between X and Y changes for each unit of Z ( Or alternatively, how much the coefficient for the relationship between Z and Y changes for each unit of X).
Cristian Ramos-Vera 's suggestion for determining moderation is correct.
To answer your question as to what your current results suggest (from the title of your query):
1. A significant covariate means only that scores on the covariate related to scores on the dependent variable (DV).
2. Significant group differences means only that, when compared at a common point on the covariate (usually the grand mean), groups had a different estimated/predicted mean score on the DV.
Thank you for the suggested method Cristian (I read it somewhere a while ago, but did not use it) and for the further clarification by David (in fact SPSS provides the common point on the covariate when I plot the results). - Reviewer asks for 'discussion' of the effect of the covariate, so I thought she/he refers to the magnitude and direction in which the covariate influences the effect of the IV on the DV. The method suggested by Cristian should do the job. Great help! Many thanks both and very best wishes!