The exact species isn't critical but from the centre of the one stair visible it's common softwood, possibly "European redwood" which is another name for Scots pine, pinus silvestris. The dark colour isn't the natural colour of the wood, it's from wood stain or due to age and wear.
You can fill holes in just about any wood in three standard ways, using:
- commercial wood fillers (of which there are a few types, including oil-based, waterbased and epoxy);
- homemade wood fillers, which are generally composed of wood dust (sanding dust or fine sawdust) mixed with some adhesive, with PVAs (both white and yellow) and epoxy being the most common glue choices;
- a solid-wood patch, sometimes referred to as a "Dutchman".
You can read a little more on homemade wood fillers in this previous Answer but given the size and location of the holes I think a commercial wood filler will work fine for you here and will certainly prove to be the easiest option for you if you're not a woodworker.
With most commercial fillers you slightly over-fill the hole to account for a small amount of shrinkage during drying, then you sand flush after the fills have hardened. With epoxy fillers there's no shrinkage so you can make the fill completely flush while the putty is still workable, saving having to sand*.
Your fills will be noticeable unless you paint over the area.
*This is good as epoxy fillers can sometimes set very hard indeed and are difficult to sand flush as the surrounding softer material tends to sand away more easily, leaving the epoxy sitting a little high unless you're very careful (it's often better to file, plane or scrape epoxy fills flush, where possible).