Use the Game Exporter (File > Game Exporter) to quickly export animation in an FBX file. You can export a list of clips into a single FBX file, or save each clip into its own FBX file.
By default, this displays the Anim Default preset, which gives you the basic FBX options for exporting animation. See Create a Game Exporter preset for information on adding your own presets.
If you have a selection set created for the animated objects, you can also select Export Object Set, then select the set itself.
The object set is maintained so you don't have to re-specify which objects to export the next time you open the file.
By default, the Start and End frame for a new clip are set to the current time range in the Time Slider. You can jog through the animation in your scene to determine where the clip you want starts and ends, then enter those frame values in the Start and End fields.
This exports the animation, based on the Filename and Path location you set.