A bailout is a situation where a white knight or other angel investor comes to the assistance of an institution under severe financial distress. Often they provide financing to allow the operation to continue for a period while they otherwise work out the difficulty.
In the case of government bailouts, several methods are possible. An infusion of cash can be used. The government can guarantee outstanding debt. The government can takeover a failing institution. Or the government can arrange for others in the industry to carry out the bailout, sometimes acquiring the weakened player, sometimes providing financing, buying some of the assets of the company, often with government guarantees on some aspects of the deal.
Recent examples include Bear Stearns, American International Group (AIG), Fannie Mae (Federal National Mortgage Association or FNMA), and Freddie Mac (Federal Home Loan Mortgage Corporation or FHLMC). Lockheed was bailed out in the '70s as was Chrysler in 1979. Long-Term Capital Management (LTCM) was bailed out in 1998.