When you heave something, you haul or lift a heavy object. You might not realize how fat your cat has become until you have to heave him out of the way to make room for yourself on the sofa.
You heave a heavy object when you lift it, like when you heave piles of wet snow out of your driveway with a shovel, and you can heave yourself out of bed in the morning, if it takes a huge effort to get yourself into a vertical position. You can even heave a heavy sigh, which simply means you sigh loudly. To vomit is also sometimes described as heaving, describing the unpleasant effort of your stomach muscles.