Acoustic Guitar
This is the instrument most people think of when they think "guitar". An acoustic guitar is hollow, and almost always has a "sound hole" - a round hole in the face of the guitar. Acoustic guitars almost always have six strings. When you strike the strings of an acoustic guitar, the instrument produces a rather loud sound. Although acoustic guitars are often associated with folk music, and "mellow" music in general, they are in truth in all styles of music, from country to blues to heavy metal.
A "classical guitar" looks quite similar to an "acoustic guitar", and is indeed still an acoustic instrument, but it has several distinct differences. Standard acoustic guitars have six strings made of steel, whereas classical guitars have six strings, three of which are nylon. This produces a sound quite different from an acoustic guitar. The neck of the guitar is also much broader on most classical guitars. I'll go out on a limb and say that unless you are interested in focusing on classical music, this style of guitar probably shouldn't be your primary choice for a first instrument.
Electric guitars have a few more bells and whistles than acoustics. Most electric guitars are not hollow, thus when you strike the strings, the sound produced is very quiet. In order to project the sound of an electric guitar, a guitar amplifier is needed. Generally, people find electric guitars to be a little more confusing than acoustic guitars - there are more knobs and buttons to deal with, and there are also a few more things that can go wrong.
Electric guitars are generally much easier to play than acoustic guitars. The strings are lighter, and easier to press down. The sore fingers that many novices experience when learning on an acoustic guitar generally aren't nearly as much of an issue when learning on electric guitar.
Electric guitars have a different role in music than acoustic guitars. Whereas acoustic guitars are often used to strum chords to many songs, electrics are used to play "guitar leads" as well as chords.
P/S: I prefer the acoustic guitar guitar for beginner and that goes to me as well, but now I stick to my electric "BLACKIE" :)