To some extent the definition depends on the type of analysis. To me, definition 1 below describes balanced and unbalanced data. However, I also found definition 2, which has more of a repeated measures flavor to it. It also fails to qualify the definition to differences in sample size, but goes directly to missing data. The Wikipedia article on Panel data has a similar description.

1. In ANOVA and DOE, a balanced design has an equal number of observations for all possible combinations of factor levels. An unbalanced design has an unequal number of observations.

2. A balanced data set is a set that contains all elements observed in all time frame. Whereas unbalanced data is a set of data where certain years, the data category is not observed.