Training is when you are fighting your own gym, to make it a higher level when you do not have a min in there.
When you have a Mon defending your turf he stays there till he gets defeated (

) but he comes back to you. He will need a potion and probably a hug.
When you have Pokemon in a gym go to the shop menu and the top right will have a shield with a number in it. The number represents how many gyms you are defending. Click this for gold coins and stardust. You can do this every 21 hours. It's worth doing when you have a few gyms if this is possible.
You can also still see you min that is in a gym in your Pokemon menu. He has a blue symbol next to him to remind you he is still at the gym. You can't upgrade him till he comes back.
Last of all if you are not on team mystic don't read this post