I got lucky and stumbled across the answer, so I'll post it in case someone
else is stuck on this one.
Go to the cell you where you want to insert the formatting (A2 in my example).
Select Format Conditional formatting:
Condition 1:
First, in the pull down select "Formula Is"
Then type =ISTEXT(A1)
Select your text display options for A2 if A1 has any text value.
Select Add to get another condition.
Second, select "Formula Is" again.
Type in =ISBLANK(A1)
Select your display options for how you want A2 to display if A1 is blank.

I was missing the "Formula Is" part and trying to do this with "true/false"
thingies.

I found this by clicking the Fx button, and clicking "information" on the
category pull down. If you have done a function task successfully before, but
can't remember how, this might be a good memory jog to actually see the
argument in the list and a brief explanation below.

"Jules" wrote:

I used to know how to do this, and can't remember. I think it is a COUNTIF
function in the first cell, together with conditional formatting in the 2nd
cell, but I can't remember how to set it up.

cell A1: any text at all entered will cause cell A2: part number to be
visible.

cell A1: empty value in cell A2 remains hidden.