This is a common question so here is a very basic guide to choosing a size of guitar to suit your child:

4-6 years-of-age: either a 1/4-size guitar - or even a soprano or tenor size ukulele

6-8-years-of-age: a half-size guitar

8-12-years-of-age: a three-quarter-size guitar

12-years-of-age to adult: a full-size guitar

Of course children are not all the same so their height and the size of their hands and fingers will vary. The above details relate to the lower end of child size percentile. Always make the effort to visit a local music shop which stocks guitars of various sizes so your child can physically hold them and place fingers on their fingerboards. Some might disagree on this but I would recommend considering a small-for-age rather than large-for-age choice simply because any child struggling to reach the necessary frets to play even simple chords will usually lose interest in the instrument. 

As to the type of starter guitar to buy well, the general consensus tends to favour nylon - or classical - rather than steel string instruments. This choice is mostly due to the fact that nylon strings are easier on the fingers and until calluses form on their fingertips children, in particular, may find steel string guitars too painful to play. However, this may not deter the highly motivated child, so don't totally ignore steel string guitars. Some children prefer them because they typically have much narrower necks than classical guitars. Of course it is possible to use nylon strings on guitars which were originally made to take steel strings (using nylon strings with "ball" ends) but unless the guitar was specifically designed to use them don't expect such a combination to work perfectly, for example you may find some nylon strings vibrate to the point that they constantly clash with the frets and create an unbearable amount of "buzzing". 

When I was a youngster children usually played the same guitar for many years because there was no cheap and easy way to sell or part-exchange it for the next size up, or an alternative model, but today thanks to the likes of eBay, there seems to be a ready market for buying and selling guitars. Well looked after instruments often change hands at close-on their original used buying price which makes the whole proposition of upgrading much more practicable for parents.