To endanger someone is to put them in a dangerous position, or to threaten their safety. It would endanger your little brother if you encouraged him to jump off the garage roof into a pile of leaves.
Texting while you drive is a good way to endanger yourself, as well as your passengers and anyone else nearby. Scientists warn that climate change will slowly begin to endanger many plant and animal species on Earth, possibly including humans. This 15th century word combines the prefix en-, "make or put in" with danger, from the Old French dangier, "power to harm."