Right, the simple version: emulsion goes on walls and ceilings, eggshell goes on wood and metal — and sometimes on walls that need to take a beating.
Emulsion is water-based, breathable, and comes in finishes from dead flat matt up to a soft mid-sheen. It's what you'll use across most of a room. Farrow & Ball's Estate Emulsion and Modern Emulsion, Little Greene's Intelligent Matt, Dulux Heritage Velvet Matt — all emulsions, all for plaster. Modern/Intelligent versions are more washable; the chalky flat ones (Estate Emulsion) look gorgeous but mark more easily.
Eggshell is a harder-wearing finish with a low-to-mid sheen — roughly 20-30% gloss, so it catches a little light without being shiny. It's designed for surfaces that get touched, knocked, and wiped: skirting, architraves, doors, window frames, radiators. Farrow & Ball Estate Eggshell, Little Greene Intelligent Eggshell, and Mylands Wood & Metal Eggshell are the go-tos. Most are water-based now, dry quickly, and don't yellow the way old oil-based eggshells did.
Here's the bit people get wrong: you can use eggshell on walls too. In a kitchen, bathroom, hallway, or anywhere with kids and dogs, a wall eggshell wipes clean and shrugs off scuffs that would ruin flat emulsion. Little Greene Intelligent Eggshell is washable enough for exactly this. The trade-off is sheen — eggshell on a wall will show every lump and bump in the plaster, so it wants a decent surface.
Practical steer: paint your ceilings and most walls in emulsion, your woodwork in eggshell. If you want the trim and walls in the same colour for that seamless modern look, match the colour across both finishes — most supported brands offer the same shade in emulsion and eggshell, so you can put Pointing on the walls in emulsion and the skirting in eggshell and it'll read as one tone. Sorted.