She bought a computer for her daughter.

Literal

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

パソコンを買ってあげた — uses あげた (polite benefactive) instead of the more casual やった. あげた sounds gentler and more generous; やった can come across as blunt or condescending. Both express doing something for another person's benefit.