Temporarily, it'd make suddenly more insightful decisions based on its existing experiences, possibly start having niggling thoughts about abstract concepts, motivations, and so on. Of course, depending on just how high the int boost goes. At int 3 or 4 we're still talking about pretty basic cognition here. I would definitely say it does not suddenly sprout the ability to understand common. It's basically the same animal, only slightly more complex in its ability to function. Think Lassie or Flipper. Still an animal, just a real clever one.
Permanently is another cup of tea. Assuming you're talking about thru the application of stat increases from levels/hd/etc - I would actually venture to say that an animal can't raise its int above 2, because that limitation is part of the animal creature type. If it's thru magical means (awaken, say) then that already has a system in place - the animal changes type to a magical beast.