Routed  timber stringboards, treads and risers are slotted in. The top section can be cut for open string.

Horizontal timber treads. This is the part you step on. Usually 22mm thick.

The vertical timber between each tread. This can be missing, it is called an open plan staircase.

A timber post that the handrail goes into on the staircase usually around 90x90mm, can be either square or bored for a newel turning to fit into

Handrails are fitted between the newel posts and is where your hands are placed. 

Spindles are fitted between the handrail and baserail. These can be square or turned.