I hear a lot of neoclassical tracks in which the guitar is doubled with another instrument such as this http://www.youtube.com/watch?v=KPZ47_qMqYA starting at about six seconds in. I was thinking a harpsichord maybe?

Sounds like a Keyboard to me.

A keyboard. I love it when people use keyboards in classical music. It's gives something new to a song.

Yep, sounds like a harpsichord, but is most likely played on a keyboard.

harpsichord. Hmm I was wondering what that instrument was when I hear it in classical music. Thanks!