Short version: you don't need special "ceiling paint" in a bottle marked as such, but you do want the flattest, deadest matt finish you can get your hands on. Here's why.
Ceilings catch raking light across their whole surface, which means any sheen throws every roller mark, lap line and bit of patched plaster straight into view. A dead-flat matt swallows all of that. Most brands' standard matt emulsion is fine — Dulux Trade Vinyl Matt is the trade workhorse for a reason, and Little Greene Intelligent Matt Emulsion or Crown matt both lay off beautifully. If you want the absolute flattest look, Farrow & Ball Estate Emulsion has a very low sheen, though it's pricier than you need for a plain white ceiling.
The "ceiling paint" you see sold separately is usually just a cheap, high-opacity flat white, often with a colour-change formula (goes on pink or blue, dries white) so you can see where you've been. Handy on big new-plaster jobs, no magic in it.
On colour: most people default to brilliant white, but a softer off-white reads far more expensively and stops that stark "strip-lighting" effect. Farrow & Ball Wevet or Strong White on a ceiling above warm walls is a lovely trick. Want it to disappear entirely? Paint the ceiling the same colour as the walls — it makes a room feel taller and more enveloping, especially in matt.
The "but what about" — bathrooms and kitchens. Steamy rooms are the one exception where a bog-standard matt can struggle with mould and wiping. Reach for a moisture-resistant kitchen-and-bathroom emulsion (Dulux, Crown and Little Greene all do one) rather than ordinary matt.
Practical bit: fresh plaster needs a mist coat first (emulsion thinned roughly 70:30 with water) before topcoats, and on stained or patchy ceilings hit the marks with Zinsser Cover Stain or BIN primer or they'll bleed straight through your white. Use a long-pile roller and keep a wet edge — ceilings show lap marks more than any wall.