Privet.

Here is a link that might be useful: Cutting hedges.

post more pix in replies to your own post ...

we need to see the whole for pruning advice ...

in addition to pix.. tell us what your goal is... i will presume its not just the shear joy of manual labor.. lol ..

it will also be helpful to know where you are ...

ken

Whatever it is, and it does look like privet, do not prune anything in the fall. Never prune in the fall. Why? You encourage the plant to produce new growth, which has two consequences. It delays dormancy because it is putting its energy into creating new growth, plus, that tender new growth will get zapped at first hard frost.

A picture of the whole hedge is shown as a link to the op, as I could only seem to post one picture. But there is no reason not to put it here also.
I am in upstate NY.
My goal is to have something resembling a hedge between us and the neighbor and to not kill it. Very modest goals.
My wife says she has heard that you have to cut the tops down every year; I thought just trimming the sides would be enough.
And if not in fall, then when?

" ....something resembling a hedge ..." It depends on how close a resemblance you want. What you have there is a classic hedge which has just not been cut this year and was not headed back as it was growing. To get it back to a formal hedge both sides and top need to but cut back. To get a really good thick hedge that one could even be cut back to half its height since it is rather sparse. To maintain a hedge once it has reached the size and density you require you cut all its new growth back.

For a less formal hedge you could just prune it like any other shrub.

Here is a link that might be useful: Hedges

Prune it in the spring. If it were a blooming shrub like a weigela (which it isn't) you would prune it after it blooms in spring. But you should prune this when the leaf buds begin to 'break'. Check out the link florauk posted.