As I know, Denmark is going to re-impose the border controls. I want to know, what exactly it means for the tourists who are on a Schengen visa? Can I visit Denmark with the Schengen visa, or I need some additional documents? I'm a Russian citizen.
From the linked article:
Because Denmark is a Schengen member, it cannot reinstate full frontier controls and the minister said the new measures would "take place within the limits of Schengen".
So yes, you can still visit Denmark with your Schengen visa and you don't need additional documents. The only thing that's extra is that you can be checked by customs at some particular border crossings. But as long as you have a valid visa, and you are not in possession of anything illegal, you should be fine.
The Schengen rules have two different aspects:
Unified visa and travel rules
Most residents or visa holders from one Schengen country can visit any other Schengen country (note that this is a generalization, check your specific case). You are allowed to cross with your existing papers.
Lack of internal border controls
Internal borders within the Schengen area are looking like strange road signs or lines on the road. For all practical purposes, you can cross without waiting in line or showing your papers. But technically you are still required to have an ID and/or a visa.
At times border controls are reintroduced. On a temporary basis, this is allowed by the Schengen rules. But that wouldn't change who is allowed to cross, just the likelihood of having to show papers.