If you see someone glower at you, you might consider glowering back, but no one likes an angry staring contest. To glower is not only to stare, it's to stare angrily, as if you're going to throttle someone.
You might already have guessed that glower isn't of Greek or Latin origin, since it doesn't ring any of those bells. Chances are that glower comes from northern Europe, where it's related to some old words that mean to "glow." It was probably in the 18th century that it took on the meaning "stare at angrily or sullenly." You can use its root, glo, to recall its meaning: a person that glowers seems to glow with anger.