!!!!!!UNDER CONSTRUCTION SO BARE THAT IN MIND!!!!!!
The music you are hearing is from Final Fantasy III

Roms and Emulators
This is where you will find the Roms for many of the classic RPG's on the NES and SNES.  A rom is a backup copy of a game cart that can be played on your computer.  You have to have a Emulator to play them so you should get that first.
NOTE: It is illegal to keep a rom for more than 24 hours if you don't own the cart for it.

Super Nintendo Emulators
ZSNES V 0.305 DOS Emulator Supports transparencies, sounds, joystick, and mouse with good GUI
SNES9X for Windows 95 Windows 95 Emulator Supports sounds and joystick with good GUI

Nintendo Emulators
 
Nesticle95 Windows 95 Emulator Supports sounds, joystick, and has superb GUI
Now on the the roms!

SNES Roms
 
Breath of Fire Capcom and Square Soft Works well with both 
Breath of Fire 2 Capcom Works well with both 
Final Fantasy 3 (JAP FF6) Square Soft Works well with both
Final Fantasy 5 Square Soft Works well with both
Final Fantasy 6 (US FF3) (Jap) Square Soft Works well with both
Lufia I don't know Works ok with DOS one but not at all with Win one
Lufia 2 I don't know Works ok with both
Ogre Battle Enix Works ok in Win but not well in DOS
Paladin's Quest Enix Works well with both
RoboTrek Enix Works well with both
NES Roms
 
Dragon Warrior Enix Works well
Dragon Warrior II Enix Works well
Dragon Warrior III Enix Works well
Dragon Warrior IV Enix Works well