As a user with the Project Manager role, go to Projects → Project list, and click Columns to the right above the project listing table.
Deselect the columns that you no longer want to see and select the columns that you want to add to the view.
Note that you can use project custom fields that you have defined in Configuration → Data → Custom fields → Project custom fields (administrative privileges required!).
Drag a square tile to the right of a column name and drop it in the position that will reflect the column's position in the project listing table.
Types of data displayed in a project tooltip are customizable as well. Click the Tooltip button and choose the data included in project tooltips the same way you customized the project listing columns.
You can return to default column and tooltip settings after you have customized them. Click Restore defaults to restore the original project listing and tooltip views.