Everything else on Kilimanjaro is a walk. Summit night is the thing people actually remember, and it helps a great deal to know what's coming.
Why it starts in the dark
Three reasons, and the romantic one is only the third. The scree is frozen solid at night and turns to sliding gravel once the sun hits it, so it's easier to walk on in the cold. Starting at midnight means you're back at the top camp by late morning and can descend to a lower camp the same day, which is the real medicine. And you reach the rim about sunrise, which is as good as it sounds.
The timings
You'll be woken around eleven at night, after maybe three hours of poor sleep, for tea and biscuits you won't want. Walking by midnight. Six to eight hours up, moving extremely slowly in a line of head torches that goes up the hill above you further than seems reasonable. From the Machame and Lemosho side you reach Stella Point at 5,756 metres; from Marangu and Rongai it's Gilman's Point at 5,681. Then forty-five minutes to an hour and a half more, along the crater rim, to Uhuru at 5,895.
The cold
Expect somewhere between minus five and minus twenty with the wind on it. You'll be wearing most of what you own. The practical problems are frozen water tubes, so carry bottles upside down or insulated, dead camera batteries, so keep them inside your jacket, and hands that stop working, so bring liner gloves under proper mitts.
What it actually feels like
Slow, cold, and oddly dreamlike. You stop thinking much. Conversation dies. There'll be a stretch, usually an hour or two before the rim, where you genuinely wonder why anyone does this. Most people feel sick at some point and a lot of people cry at the top, including people who would tell you they're not the type.
At the top
Less time than you imagine. Fifteen or twenty minutes, long enough for photographs at the signboard and a look at the glaciers, and then your guide will want to move you down, because 5,895 metres is no place to stand around. If it's busy there's a queue for the sign. Nobody tells you that part.
The descent nobody warns you about
Down to the top camp in a couple of hours, often half-skiing the scree, then an hour's rest and a meal, then another three or four hours down to a lower camp. That's a fifteen-hour day on three hours' sleep, and it's where knees give out. Walking poles earn their keep here more than anywhere else.
If you turn around
Some people do, and the guide's call is final. It isn't a failure, and it doesn't mean you were unfit. It means your body said no on that night. The mountain is still there.