The portion a company paid in excess to fair market value when purchasing another company.

Let's say Company A purchased Company B for $100m when company B's fair market value is $75m, leaving company A paying $25m in goodwill.

Why would a company pay goodwill?

Goodwill is an intangible asset which includes items such as brand awareness, intellectual property, proprietary property. There are many times where a company's intangible assets are very valuable and thus is acceptable to pay above book value of fair market value for a company.