Guitar Hero III is the continuation of the Guitar Hero series, originally developed by Harmonix. When publishers and developers for the title changed hands, the Guitar Hero series landed in the hands of Neversoft (the developers of Guitar Hero III), while Harmonix moved on to develop Rock Band. Both Guitar Hero and Rock Band are an evolution of the gameplay of Harmonix' earlier titles Frequency and Amplitude, with the biggest change being the addition of peripherals to improve the controls.
Guitar Hero is best known for its guitar peripherals, and Guitar Hero III doesn't change things much. This being the first game in the series developed by Neversoft, they focused more on developing a strong Guitar Hero title rather than changing the formula. The major new change is that the official bundled guitar controller is now wireless.
The basic gameplay of the Guitar Hero series involves pressing the colored fret buttons on the guitar controller's neck, and hitting the strum bar up or down to "play" the notes as they reach the bottom of the screen. The fret buttons can be pressed and held until the proper time comes to hit the strum bar. They may also be held for consecutive notes on the same fret, and some notes will have to be held for sustained notes (indicated by a line heading upward from the note).
A gauge on the side of the screen (called the Rock Meter) tells you how well you're doing. If the meter falls too low (it will flash red when you're in trouble, but starts at green and moves through yellow to red), the music stops and you can either start the song over or choose another song (or practice).
If you are new to music games in general, you may want to start with the Easy difficulty. If you're familiar with music games and simply new to the Guitar Hero series, you may be comfortable starting with the Medium difficulty (you can always drop down to Easy if you're having trouble getting used to the guitar).