To export from Blender, make sure your content has been weighted to all the bones, has an armature modifier attached that is linked to the rig, and then choose “File/Export/FBX”.
If you are exporting only the selected items, make sure you have selected both the mesh(es) and Armature that they are attached to.
Fill in the FBX options as below:
This is crucial - you must choose “X Axis” for the Primary Bone Axis. And “-Y Axis” fro the Secondary Bone Axis:
When importing in Unity, you will usually need to set the FBX scale to 100.