Magento – Understanding data_index Column in catalogsearch_fulltext Table

The checkbox on each line in the grids is a way to mark if the product listed on that line is in relation with the product you are creating/editing (up-sell, cross-sell, related).
In general a grid in Magento has the following elements.

1. The first row represents the column headers. These headers give meaning to the columns.
2. The second row contains the filter options. It allows you to filter by any column that supports filtering. This row is not mandatory. It can be omitted on some grids (see the Reports grids for example).
3. The rest of the rows represent the entities listed in the grid.

In the case of the relations tabs (and others) the dropdown with the values Any, Yes, No represents a way to filter the grid (because it's on the filter row of the grid).
If you select Any it means that there is no filter on the grid for the column with checkboxes.
If you select Yes and press Search you will see in the grid only the rows that have the checkbox checked. (Of course the rest of the filters apply if you have any). If you select No and press Search you will see in the grid only the rows that have the checbox uncheched.
It's somehow similar to this. The difference is that in this case there are checkboxes not radio buttons. [EDIT]
The purpose of the checkbox above the Any/Yes/No filter is to check/uncheck easier all visible checkboxes.

The basic answer is "Magento search doesn't work well in either mode, get another search engine system." I have done all the above except truncating the query and result tables. If you want to waste a lot of time like I wasted a lot of time trying to make a minimal function search system work, feel free to pursue anything you want, but Sisyphus didn't have much return on effort either.

Modules are available for Sphynx, Lucene, Solr. In this instance, Free is lost sales so go for one that implements one of the above and doesn't try to patch the stock Magento search.

Anything 3rd party will give you 80% relevance or better and actually let you sort by relevance.

Stock Magento is in the 20-40% range for return results and that's as good as it gets. Also the random return is based on product order, not relevance order for LIKE so 500 returns probably means the relevant ones are 2/3 the way through.

When I tried it, the Mysql "Fulltext" never added any actual relevance (separate from relevance rating) to the search and defaults to not searching for words that are shorter than 4 characters (no stop-word functions, we'll just ignore everything so that actual 3 character part name NEVER is found).