Okay, yesterday I explained what a hand is and that it's used to measure the height of a horse.
The obvious next question is how to measure a horse. In the old days, of course, horse heights would have been estimated. Horses have also become bigger over time...back in Medieval times, most horses were 12-14 hands - we would call them ponies today. A buyer would, in any case, most likely have the animal there to look at. Approximation would have been quite acceptable. If your story is set anywhere prior to the mid nineteenth century - or equivalent tech level - you might not even bother discussing the height of a horse in any detail.
Today, we have height restricted classes for ponies, newspapers, and the internet - so we need to measure horses quite a bit more accurately.
The downside is that the skill of estimating a horse's height has been lost, particularly over the last thirty years or so. I was taught to do it by the simple expedient of drilling me - with rewards for the most accurate guesses. Most horse people today can't estimate height and they almost always guess it at least a hand too high.
The correct, official way of measuring a horse is to use a measuring stick. This consists of a vertical stick - either wood or aluminum, and a horizontal wooden or plastic bar. Most modern measuring sticks are marked in hands and inches on one side and centimeters on the other. Aluminum sticks are preferred because they can telescope for easier storage. This is a typical modern aluminum measuring stick.
It takes two people. The horse is led out onto a flat surface and one person holds the horse with its head slightly down. The other places the stick next to the horse's shoulder and slides the bar down until it rests on the highest part of the horse's withers - the small bump at the top of the shoulder. For official measurements - particularly for "pony cards," which are considered proof of height for international height restricted classes - it's required to remove the shoes. Most of the time, you just allow half an inch for shoes.
You can also measure horses with a tape, but this takes three people (One to hold the horse, two to hold each end of the tape). The tapes are trickier to read and affected by wind - but if you just want a very rough estimate, they're fine. They're also inexpensive. I personally find that a trained person can get just as good a guess by eyeballing the horse, or standing up next to it.