She bought a computer for her daughter.

Literal

She [topic-は] daughter [purpose-のために] personal-computer [object-を] bought gave.

パソコンを買ってやった — てやる is the casual/downward benefactive, used when doing something for someone of lower status (child, pet). パソコン is the standard abbreviation of パーソナルコンピュータ ('personal computer').