A user will select one value from a popdown menu. The selected value should determine whether another entry field will be shown (required) or hidden (superfluous). I wonder if this should be controlled by a script. However, I can't think of how a script would be activated automatically in response to a user input (in this case, the entering of a value from the popdown menu). Any help would be much appreciated.

You don't need a script. Make a relationship based on the first field. Put the second field in a portal based on this relationship. If the relationship is valid the portal will show and so will the field, if the relationship is not valid the portal and field will not show. Make the lines for the portal transparent.

Also, using one calc field -

Create a calculation field that returns exactly what's in the shown/hidden field only when the popup selection wants it hidden. Line up that calc field directly on top of the shown/hidden field, and make its text color match the background color.

This method can also hide the content of a field, or any text, that's already in a portal.