Put simply, the term 'Russian Oligarch' refers to Russian businessmen who rapidly accrued large amounts of wealth through the purchasing of Soviet assets after the dissolution of the Soviet Union (1991). Effectively business magnates, they saw the opportunity of the Soviet Union's vast production assets (factories, mines etc.) and bought them off the state.

For those who don't know, the transition from communism to capitalism involved the sale of most state-owned industry, due to the nature of the transition there was a huge amount of denationalisation which needed to occur and as a result most state-industry was effectively given away. Newly freed from communism, those who are today often known in the media as 'Oligarchs' are the men who took ownership of these industries. I won't talk about it in detail as it's a slight meander but many of these Oligarchs amassed such industrial wealth through corruption, bribery and other dishonest means. Ultimately many of these men became Billionaires essentially dividing up the Soviet state amongst themselves and reaping the rewards. The term 'Oligarch' is used somewhat loosely in the media sometimes just referring to very wealthy individuals from former Soviet countries.

TLDR: Men who made fortunes through buying up state-owned industry when the Soviet union dissolved; they became rich through owning large factories and other economic facilities

EDIT: Added some punctuation and Soviet dissolution date

I am terrible at linking, but Karen Dawisha recently published a book called "Putins Kleptocracy" which is extremely interesting on this point. Here is an extract from Anne Applebaums review in the NYROB.

Begins:

Using this mass of evidence, Dawisha nevertheless argues that the KGB’s return to power begins not in 2000, when Putin became president, but in the late 1980s. At that time, the then leaders of the KGB, who distrusted Gorbachev, began transferring money that belonged to the Soviet Communist Party out of the Soviet Union and into offshore accounts tended by Swiss or British bankers. At least initially, these transfers took place with the Party’s knowledge. In August 1990, the Central Committee called for measures to protect the Party’s “economic interests,” including the construction of an “invisible” structure, accessible only to “a very narrow circle of people.” KGB operatives who already had experience with managing foreign bank accounts—they’d been funding foreign Communist parties for decades—were put in charge.

           Ends. 

The fundamental point is that there were gigantic liquid assets lose in the cccp at the fall. Where that money went, and who ended up managing it is a big part of the answer to your question.