This iconic track first appeared at the title screen and throughout the overworld of the original Legend of Zelda and has since featured at some point in every Zelda game since, with the notable exception of Ocarina of Time.
It's composed by Kōji Kondō, who has worked on all the Nintendo-developed Zelda games. He is also the chief composer for the Super Mario franchise and is responsible for the beloved Mario theme. Kōji's other works include Duck Hunt and Star Fox 64 (known as Lylat Wars to Australians).
The version of the Zelda Theme above is as performed by Zack Kim on dual guitars. Another very excellent electric guitar arrangement can be heard by clicking here.