When something is sheer, it's a little see-through. So e.g. an opaque blue lipstick would turn your lips completely blue without the color of your lips being visible, while a sheer blue lipstick would also turn your lips a shade of blue, but you'd still be able to see the original color of your lips underneath.
If something "sheers down", it means that it loses its opacity and becomes see-through over time.
A stain element, in my experience, means that the lipstick will stain your lips, meaning that it will leave some of its color on your lips even after you wipe it off. If you had a pink stain lipstick, it could stain your lips and turn them much pinker than normal for a while, even after removing it.
Hope that helps! :)