The shock absorber boot (or dust boot) covers the polished chrome rod of the shock. It acts as a shield against road grit, sand, and water. If dirt gets onto the rod, it will tear the internal shaft seal, causing the shock absorber to leak oil and fail.
Torn boots should be replaced promptly. While the car remains safe to drive, the exposed shock shaft will pit and rust, leading to premature oil leaks and a rough ride.