What a stair riser does
Structurally, very little: the riser covers the front of the step instead of carrying anything. Visually it does a great deal, because it is the surface seen from below and from across a room, and there is one of them for every step.
That repetition is why the riser decides how a staircase reads more than the tread does. A flight of stairs seen from a hallway is mostly risers.
Matching the tread
The default is to match the riser to the tread and the nosing, so each step reads as one piece of material. It is the quiet option and the one that suits a staircase meant to blend into a floor.
The alternative is a contrast, most often a painted white riser under a wood tread, which makes each step legible from a distance and lightens the whole flight. Both are deliberate looks, and the stair nose page covers the edge that sits between them.
Risers supplied as boards
Risers come as boards rather than as fixed sizes, because step heights vary between staircases and often between steps in the same one. The board is cut to the height and width each step needs.
That changes how quantities are worked out: what is being bought is a length of material and not a set of parts, and the number of boards depends on how many risers can be taken from each one.
Step heights vary
Step heights are rarely identical through a flight, particularly at the top and bottom and in older houses where the staircase has moved.
That is why risers are supplied as boards to be cut rather than as finished parts, and it is also why measuring every step rather than one of them is the habit that avoids a shortfall.
Open and closed staircases
A closed staircase has a riser at every step. An open one has none at all, with daylight through the gaps, and needs no riser material.
Some staircases are open on the side but closed at the back, which still needs a riser at every step while changing what the tread needs at its exposed end. That combination is worth identifying before ordering, since it affects both lists.