Stair terminology

Stairwell: The opening, containing an indoor stairway.
Flight: An uninterrupted series of steps.
Balustrade: The system of handrail and balusters that prevents people from falling.
Spandrel: The (usually triangular) space underneath the stairs
Soffit: Underside of stair
Apron: Vertical perimeter around the stairwell

Utility Stair: Stair not used for day to day use, e.g. escape, access, maintenance, or sometimes loft conversions
Private Stair: used in only one dwelling
General access stair: used for more than one dwelling in private building, e.g. shared stair in a block of flats

Geometrical Stair: An elegant and highly desired staircase using curved strings and handrails (rather than a standard stair)
Spiral Stair: Simple, efficient space design, can be unconfrotable to walk up
Eliptical Stair: Geometrical Stair curved by an ellipse (rather than an arc)
Straight stair: Straight flight
Quarter/half turn, Multiple straight flights, that change direction through the stair, on either winders of landing
Open Stair: stair that doesn’t have risers. (rather than the usual ‘closed stair’ which does have risers)

Tread: The horizontal part of the stairs that is stepped on. Tread types include

Flyers: Straight Treads
Winders: Angled Treads
Curved Treads: Treads with curved risers, attractove for high class staircases
Scroll / Bullnose Step: Decorative curved first step providing extra space for a scroll or Newel post and often

Riser: The vertical parts between each tread on the stair.
Nosing: the part of the tread that protrudes beyond the riser (or string on a cut string staircase).
Landing: A flat platform at the top, bottom or between flights, should be as wide as the stair in both axes.
String: Supports treads and risers on either side of the stair, variants include:

Closed String: Treads and risers are recessed into the string, typically lower cost.
Cut String: Cut underneath treads/risers, traditional and higher quality stairs.
Rough String: Used for support but is not seen in finished stair

Pitch / rake: Angle on stair or stair parts (e.g. handrail)
Pitch / nosing line: line that runs across stair nosing’s (parallel to the pitch/rake)
String Capping: Beading that site on top of wall strings, provildes detail and hides unevenness in wall
Scotia: Moulding under treads/nosings on cut string staricases

Spindles / Balusters: (typically) vertical posts under handrail which prevent falling through the gap, often decorative
Newel Post: Larger baluster / post used to anchor the handrail.
Handrail: Rail mounted on the wall or on top of the guarding / balustrade

Geometrical Handrails: A continuous curved handrail that run without break, caps or newel posts along a (typically geometrical) staircase. Geometrical handrails are by far the most desired type of handrail, but can be more expensive than straight handrail mounted into newel posts.
Scroll Terminal / Volute: The decorative handrail piece at the start of the stair that curls around in a spiral, high quality scroll terminals are pitched low to provide a more attractive look.
Wreaths: The sections of double curvature handrail, often also twisting around corners to form a continuous handrail. Wreaths are carved out of solid timber using precise geometrical techniques and highly skilled, specialized craftsmen.
Cap: Elegant but simple termination of a handrail above a newel post
Facemold: Geometrical form used in the construction of Wreathed handrail sections
Easing / Ramps / knees: Curved handrail in a single axis, generally easing = horizontal, ramps / knees vertical.
Swan neck: An vertical handrail component that joins a sloped handrail to a higher handrail, simple to make but typically undesirable.
Profile: The cross section of a handrail

Finial: A decorative cap to the top of a newel post or underside of a side mounted spindle.
(Tread) Bracket: A decorative design placed on the cut stringer.
Runner: Carpeting that runs down the middle of the stairs.