Top box and bottom box are aggregation methods for Likert scale (rating) data. They work by simply calculating the percentage of respondents that gave a rating at the top or bottom of the rating scale.
Top 2 box is a common approach to analysing Likert data. For a top 2 box calculation we would present the percentage of respondents who gave either of the top 2 ratings from the scale.
For example, if our rating scale is 1 - 10, then top 2 box would be the percentage of respondents that gave a 9 or 10. Conversely, bottom 3 box would be the percentage of respondents that gave a 1, 2 or 3.