Corner points in fontforge (seemingly also known as "coins", shown as squares in the UI), can have zero, one, or two handles to control the direction of the outgoing lines. This fact is nicely explained in the manual, but I cannot find any hints on how to control the number of handles.
How do I change the number of Bézier handles of a corner point?
In my specific case I have a straight line between two corners, and I would like to change this line to "bulge outwards" a bit. I tried to add a "curve point" between the two corners, but this gives me a strange curve point with no handles, which seems to be functionally identical to just another corner.
The simplest way to make a line bulge is to grab the line and drag (ensure the surrounding corner points are not selected first). This will immediately turn it from a line into a curve, and create control points. If you want to turn it back into a straight line, select the surrounding corner points then right-click the line and choose "Make Line".
You could equally add a curve point between the corners (CTRL-ALT-click where you want to add) and then merge it back out of existence using CTRL-M. This will leave the line as is, but with control points available.
Note that control points won't display unless you have selected an adjacent corner point or curve point. You can make them always display by selecting View>Show>Control Points (Always).
Very occasionally you may encounter what looks like a line but is actually a curve in which the control points coincide exactly with the corner points. The above methods will work fine in this situation too.