Flippant answer: Russia does NOT, in fact, have a Prime Minister :)
More specifically, while Medvedev is sometimes colloquially called "PM" by equivalence, the term "Prime Minister" is not mentioned anywhere in Russian constitution. The official title of the post is The "Chairman of the Government of the Russian Federation (Russian: Председатель Правительства Российской Федерации)".
Having said that, there are different situations in different countries.
Specifically, the situation in Russia can be characterized in layman's terms as:
The best analogy I can think of is a division of the corporate roles of Chairman of the Board vs. CEO. The former is responsible for heading the strategic direction of the company, and typically works with the Board to choose and hire a CEO who in turn is responsible for actual day to day executive management of the company.
P.S. Of course, in Russia, this separation is purely nominal, as - despite the cosmetic castling, Putin remained the "CEO" role holder despite being PM for Medvedev's President in between his own two presidential terms.
As far as examples, as noted, most corporations have this dual role. Some other states have it as well, but in different forms - for example, while Israel has both a PM and a President, the President holds significantly less real political power.