Enmity is a mass noun that speaks of a state of hatred or ill will between two people or groups. This opposition is acted upon and usually mutual. The plural is enmities.
Animosity means a feeling of hatred or ill will. It is also a mass noun. The plural is animosities.
A related word hostility also means a long-held mutual hatred, but it is usually used to describe the actions taken in response to this feeling. It can also literally mean acts of warfare. The plural is hostilities.
To a certain extent these words can be used interchangeably. However, if you want to make a distinction, some dogs feel animosity toward cats; they have enmity against cats, and commit acts of hostility when they see cats.
Quarrel erupted between two youngsters during a temple festival about a month ago. The incident sowed seeds of enmity between them and ultimately resulted in a murder. [The Hindu]
Decades earlier, Arthur, leader of the Britons, had asked his wizard Merlin to conjure the dragon mist so that old enmities were forgotten and peace settled on the land. [Sydney Morning Herald]
He has lost two major elections, parliamentary and presidential in 2000 and 2005 but in recent years he became popular among supporters of defeated presidential candidates in 2009, because of his long history of animosity with Ahmadinejad and his open criticism of the government suppressive policy toward green movement. [Politico]
Cultural animosities would pose potential obstacles to a union between groups that are not known to have even exchanged emissaries, U.S. officials said. [The Washington Post]
The mood in the Capitol has shifted from skepticism two years ago, to very cautious optimism last year, to outright hostility. [The Kansas City Star]
With political hostilities suspended for the day, hundreds of supporters of Hasina’s Awami League party staged a victory procession at Motijheel, the main commercial district in the capital Dhaka. [Bangkok Post]