A closed staircase, also known as "Boxed Stairs" will have a wall on either side of the staircase. The stair treads and risers will but up against the staircase stringers on either side.
Open one side stairs
A one side open staircase will have one side of the treads and risers adjoining the wall, where as the other side of the treads will slightly extend (which is the nosing and return) over the stringer. Spindles or pickets will be inserted into the stair treads open side of the staircase supporting the handrail.
Open both sides stairs
A both sides open staircase will have both sides of the treads slightly extending over the stringers (which is the nosing and return). Therefore with a both sides open staircase you will have spindles or pickets on either side inserted into the stair treads supporting the two handrails.
Open riser stairs
An open riser staircase only has treads. There are no risers. You will be able to look straight through the staircase. This is great for allowing light to enter an area, where as in a conventional staircase the risers would have blocked the light. The underside of the treads must obviously be in a finished condition. One caveat is that you lose "Under the stairs" storage.
Spiral or circular stairs
A spiral or circular staircase will typically turn 90 degrees from top to bottom (or from floor level to floor level). One side is usually open and the other closed. Along with a "First step" usually curved on the open side, you will have a combination of rectangular as well as "Pie shaped" treads which makes up the 90 degree angle between levels of the home.
Stairs with Landings
A staircase with a landing will typically turn 90 degrees from top to bottom (or from floor level to floor level). All of the treads will be a rectangular shape. The landing which is typically about 10 to 12 square feet in area will allow for the 90 degree turn in the staircase. Landings can either be finished in regular hardwood flooring planks or boards, or a solid hardwood piece cut to fit.
Stairs with triangle and Trapezoid treads
A staircase that has triangular and trapezoid shaped treads will also generally make up a 90 degree turn in the staircase from top to bottom (or level to level). This type of staircase will have a triangle then a trapezoid followed by an additional triangle shaped tread. These odd shaped treads are cut out of solid hardwood to fit