The Earth-Kzin Wars were a series of four conflicts between Earth forces and the Kzinti in the mid-to-late 21st century.
Although the Kzinti had opportunities to taste "Human meat" during these conflicts, ultimately the wars ended badly for the Kzinti with a decisive victory for Earth.
In the end, Earth imposed many restrictions upon the Kzinti, through the provisions made by the Treaty of Sirius. These restrictions included the limiting of their space fleet to a few police vessels. (TAS: "The Slaver Weapon")
Little was revealed of the details of the wars, but the date of the last of the four wars were stated as being "two hundred years ago" from 2269, placing it sometime in the late 21st century.
Given the timeframe, this is possibly the earliest major conflict between Earth and another race.