Solid Hardwood, Engineered and Laminate Flooring - question about oak stair treads

I am getting ready to put some stair treads in. Why do I see a wood piece along the wall at the sides of stairs. looks almost like a baseboard but usually about 10" wide running the length of stairs.

That would be the skirt board. A baseboard of sorts for the stairs.

any particular width that the skirt board should be? like how far above the nose of the tread? thanks

I dont believe there is a 'standard' distance off the nose of the tread - more like what looks most desireable to you.
What Ive seen is about 1.5-2"...but it might depent on how 'open' the staircase is...an adjacent wall...etc.
Where/how does that 'skirting' fit now..and how will it look when you install your treads?
Are you also installing full risers (I presume) to match the treads? If so, then your original 'skirting' might fit back into place after the install of new treads and risers!?!?

There is not existing skirting. I am installing risers and treads. The skirt board will not have to me notched either. I was figuring on installing it first and then cutting the risers and tread to fit. the 1 1/2 to 2 inches is good though. thanks

Ok..
Is there skirting attached to the wall (" Why do I see a wood piece along the wall at the sides of stairs.)" ..
.
Or Not "(There is not existing skirting.)"
.
If there isnt any skirting, do you want skirting...?
You dont have too!
Might look cleaner/more 'open' without it.
Depends on the staircase and area...(imo).