A half nelson is a pressure hold which is most often used in wrestling. It is similar to a full nelson, however, the attacker only locks in one arm instead of both. A half nelson is most often applied by getting behind the victim, and passing your arm under his corresponding arm (e.g. your right arm goes under his right arm), and reaching up and around to grab the back of the victim's head. This makes it almost impossible for the victim to counter-attack the attacker (since the attacker is behind the victim), and the attacker has limited control of the movement of the victim. It can also very difficult to break free from this hold unless the victim is significantly stronger than the attacker.
One advantage of a half nelson over a full nelson is that it leaves the attacker with one hand free to cause damage or inflict pain elsewhere on the victim. (This is not an inconsiderable advantage).
Disadvantages include less actual control over the victim with a half nelson, and it is rather easier to break free from a half nelson.
There is some doubt as to the origin of this word, but the one I've come across most often is that the term (and actually all the nelson varieties: full, half, three quarters) is named after Nelson, a town in Lancashire, England, which was popular for its wrestling matches. The town of Nelson was previously called Marston, but changed its name to Nelson after an inn located there (The Lord Nelson Inn... duh!) which was in turn named after the English naval hero, Admiral Lord Horatio Nelson.
The other (more obvious) origin of the term, is that it was named after a wrestler who was skilled in applying the maneuver. I haven't been able to find any information about this wrestler, and he is almost universally anonymous except for the obscure reference to his name. Therefore, I am inclined to believe the former etymology.
But hey, that's just me.