Hunter (full hunter): a solid front lid protects the crystal. Pressing the crown or pendant releases a spring catch. The winding stem is usually at 3 o'clock relative to the dial, so a hunter movement is laid out differently from an open-face one.
Half-hunter (demi-hunter): the lid has a central glass window and a chapter ring on its outer surface, so the approximate time can be read with the lid closed.
Open-face: no front cover. The crown is normally at 12 o'clock, with the seconds dial at 6.
Why it matters for identification: a hunter movement placed in an open-face case, or the other way round, puts the seconds dial in the wrong place relative to the crown. It is a common sign of a recased or composite watch.
Care: close a hunter lid gently by pressing the crown in, so the catch does not strike the case. Snapping it shut wears the catch and the hinge.