To the right of the search box on the web app's record editor, you will see a button with three red lines. This button will open the column setup window.

In this window, you can enable and disable the fields that you wish to have displayed in the table. You can also click and drag on the three horizontal lines to move the ordering of the columns in the table.

Some system fields (record_keys and record_series) are only populated in Tasks and Issues apps. These columns are hidden by default in other apps, but are available to be displayed using the 'Show hidden column' button.

Resetting your columns or hiding it by clicking on the column name and selecting 'Hide Field' will be hide them again.

Reset columns to defaults using the Column Setup interface:

Hide individual columns by clicking on the column name and select 'Hide Field'.

When you run an export using the Download Data button on the record editor, the exported data file will only contain the columns that are visible in the data table. However, there are a few columns that are deemed to be critical: the record ID columns, location data (geometry and latitude and longitude), child (repeatable) data, and create/update data. Since these are necessary columns for the importer, they are always included in the export data file.