Well this really has a lot to do with your musical tastes and influences.
An Acoustic guitar if steel strung (steel strings) could be suitable for Folk songs, Country, Pop etc in fact it can crop up in almost any area (genre) of music and so is a very popular choice of guitar.
An Electro or Semi Acoustic guitar (this has facility to be plugged into an amplifier to make the sound a lot louder) can be used as above.
A Nylon strung Acoustic guitar is usually used for Spanish Classical, Baroque, Classical playing and has a mellow sound, often played with fingers and no pick. It seems to be the popular choice for children as the strings can be easier to deal with for small hands/fingers but the fretboard is quite wide. It depends on what type of music your child is interested in learning and your child’s size as to what type of guitar is best for them.
An Electric guitar is a very versatile instrument given the fact that it can be easier to learn and play than acoustic and you can connect effects pedals (stomp boxes) as they are also known to create some fantastic sounds,this together with the amplifier used can give some pretty cool and weird, at times, sounds.