Fragility is a state of being delicate or breakable. The fragility of your grandmother's juice glasses might make you nervous about taking even one sip.
An object's fragility makes it likely to get broken or damaged, and a person's fragility means that he's not physically strong. New parents are often worried about the fragility of their newborn baby, and being sick can emphasize a person's fragility. Another way to use the word is to mean "vulnerable" or "sensitive." When something's delicate or frail, you can call it fragile, and both words stem from the Latin root fragilis, "brittle or easily broken."