I often get students who sign up for lessons through a form and they specify they want “classical guitar lessons”. After I meet them I find out that what they really mean is they want “formal training” in guitar, that is, they want to learn to read guitar music notation and go through a structured curriculum that covers all of the basics of guitar technique, and the associated music theory. Bravo for them. And that is exactly what they will get. (I teach a more relaxed curriculum for those that want that too). However, say “classical guitar lessons” to a guitar teacher and they will assume you want to learn to play a specific type of guitar called “the classical guitar”. So what is a “classical guitar”? It is a specific type of acoustic guitar with a wide neck (and wider spaces between the strings to make finger-picking easier). Instead of the steel strings found on electric guitars and most acoustic guitars, the 3 high strings are made of nylon and the 3 low strings are made of silk with steel wound around it. This makes the strings easy on the fingernails of the right hand, which are used to strike the strings. While steel string guitars are usually played with a pick (though they can also be played with finger-picks or fingertips), the classical guitar is played with the fingernails.
The music played on this style of guitar is usually classical music. But that doesn’t necessarily mean only the music of dead guys who wore wigs. There are plenty of interesting modern composers writing music for the classical guitar (e.g. Stuart Weber, Leo Brouwer). It is often used to play latin styles of music and some jazz guitarists prefer the sound of the classical guitar (e.g. Earl Klugh, Charlie Byrd). You may have heard the classic guitar without necessarily realizing just what it was, if you have ever heard recordings by Andres Segovia (who popularized it), John Williams, Julian Bream, Liona Boyd, Christopher Parkening, David Tanenbaum and others.