For "my love", you can say "mon amour", literally. Mon ch�ri, to a man, or ma ch�rie, to a woman, are less "dramatic" though! lol
Ami/amie primarily mean friend, not partner, although as Fred said, it can be used to mean partner in some cases (when the couple is a bit older, for example, and saying "mon copain / ma copine would sound a bit childish)
Girlfriend is copine, boyfriend is copain. However, over a certain age (say...30?), it sounds a bit stupid, so you would revert to "mon ami/e", as said above.
Petit ami / petite amie are both quite old fashioned expression, although still in use, especially in novels.