A federal city is a city which is a separate entity. It has a status of both: an inhabited locality and a federal subject.
There are three federal cities in Russia: Moscow, Saint Petersburg and Sevastopol in the disputed region of Crimea.
All three cities have federal importance and federal power. Parts of those cities are highly important government/military areas with very restricted settlements rights.
The federal power is seen most in federal representation. A federal city has its own members in the Federal Council, its own legally elected legislature and elected/appointed executive authority. A federal city can create its own laws, has its own taxation power and the power to subdivide the city in smaller municipalities. It has many special privileges which ordinary cities don't.