The stomach is the organ of your body — kind of a big sac — that digests food. We also say people have a stomach for something unpleasant when they can handle it.
Without a stomach, you'd be in trouble: that's where your food gets digested and turned into useful material your body can use for energy. If you ever got punched in the stomach or had a stomach ache, it can hurt a lot. Also, stomach can mean having a hunger, or at least tolerance, for something. If you can't stomach watching football, then you can't stand watching it. If you have a stomach for something, you can swallow it.