In most cases that I've seen in writing, "loosing" is a misspelling or misuse, where "losing" is the proper term. To "lose" can mean either to stop having something you once had (as in "I lost my pencil" or "I lost my sanity"), or it can mean the opposite of winning.
Loosing is a word that is rarely used (at least properly) and refers to the verb "to let loose," as in "to set free, either temporarily or permanently." It would apply to loosing an animal, either setting it free or for hunting or chase ("let loose the dogs of war" Shakespeare).
Loosening is not the same as loosing. We loosen something when we make it less tight, but we don't release it entirely. Now, if an animal is tied up, loosening it's tie or reins might be a way of loosing the animal, but we'd probably be better off releasing the reins, lest the loose animal get tangled!