They went to a largely capitalist economic system, and started to choose their leaders with democratic elections (or at least elections that had the veneer of being democratic. They also tried to project a more peaceful image -- convincing many that they were interested in cooperation, not confrontation. They stopped most of their anti-Western rhetoric, and even joined the G-8 economic group.
And they were pretty successful in their efforts to change their image over the last couple of decades. Many in the West began to believe Russia was serious in its desire to join with the West in a desire to promote peace and prosperity, and that they should no longer be viewed with fear and trepidation.
The recent Winter Olympics in Sochi was Russia's opportunity to show off the "new" and changed Russia. And that effort was largely successful. While many disapproved of Russia's stand on homosexuality, most saw through the games the image Russia wanted to project -- the image of a friendly people who just wanted to take their rightful place among the world's nations.
Unfortunately, with one single action (the invasion of the Crimea by Russian troops), Russia has managed to destroy two decades of hard work in rehabilitating their reputation. They say they acted only to protect Russians living in the Crimea ( a section of Ukraine with a large Russian population). But the fact is that they have invaded a sovereign nation -- bringing back visions of the old aggressive and expansionist Russia.
Will Russia withdraw its troops from the Crimea once the situation has settled down? Or will they annex it (and possibly other parts of Ukraine) as an autonomous Russian-protected (and controlled) country? I suspect it is the latter, but only the future can answer that question. Either way, they have done some serious damage to the rehabilitated reputation they worked hard for two decades to establish.