A legally declared inability to pay back debts.
A bankrupt person loses everything except some basic things they own, but all the debt will go away. They receive a bad credit record and may not be able to borrow money again for years.
A bankrupt company gets protection from people it owes money to (so they cannot destroy all of the physical capital and goodwill by breaking it apart and moving it away). This provides more time for the business to work out a good solution.