A conical burr grinder turns a cone inside a matching ring. The geometry is compact, quiet and forgiving, which is why it is the most common shape in home grinders — and usually the right place to start.
Beans fall into the gap between a cone and the ring around it. As the cone turns they are pulled downward and cut progressively smaller until they escape at the bottom.
Because gravity does much of the feeding, a conical set can run slower and cooler than a comparable flat one. Less heat means less of the aromatic loss that comes with a hot grind.
They cover a wide range without complaint — the same grinder handles French press in the morning and something finer at the weekend.
They are quieter, they take up less counter height, and they are more forgiving of an imprecise setting. If you are moving up from a blade grinder or from pre-ground coffee, this is where the jump in cup quality is largest per dollar.
Where they give ground is espresso resolution: a conical produces a slightly wider spread of particle sizes than a comparable flat set.
Choose conical if you brew more than one way, if noise matters, or if you would rather not chase a dial every morning.
Choose flat if espresso is your first priority and you weigh every shot.
Burr diameter matters more than shape at the margins: a large conical will out-perform a small flat set. Compare sizes before you compare geometries.