What does it mean when one model fit the data better than does a comptitor model?
You need a criteria for saying whats better (i.e. a risk function). You say estimator A is better than estimator B with respect to a given risk function if the risk of estimator A is better than estimator B. 

In the case of estimation, the risk is often the mean square error, i.e. if $\hat{X}$ is your estimator and $X$ is the thing you're trying to estimate, $R(\hat{X}) = E[\lVert X - \hat{X} \rVert_2^2]$ where $\lVert \cdot \rVert_2$ is the 2-norm. Then, you say estimator A is better than estimator B if it has lower mean square error.

When working in the mean square error framework for deterministic parameters, you have the notion of efficient estimators, i.e. ones which meet the Cramer-Rao lower bound, which are the best you can do in some sense (as described on the wiki article).