Hunts are a massive side quest in Final fantasy XII. You are given your very first hunt as part of the main storyline. But after that, the hunts are strictly optional.
In most towns and cities, you will find a board, with the available hnts on them. To begin a hunt, simply go to the board, and select a hunt from the list. It will then give you a brief synopsis of the hunt, and tell you of the location of the "Petitioner". The Petitioner is the person who has posted the hunt on the board, and in most cases the one who rewards you.
In order to make the hunt official (As well as make the mark appear), you need to speak to the petitioner personally, and accept the hunt from them. Once the hunt has been accepted, you will see: "Let teh Hunt Begin" under the marks name. This will let you know you can now go after the mark.
The petitioner will provide you clues as to the whereabouts of the mark you need to hunt, but they are not always great with the details. Some marks require certain things be done, before they appear. This will be mentioned by the petitioner, so pay close attention to what the person is saying. And these conditions can be anything. A couple of examples are one mark requires a female only group enter the area, before it appears, while another requires characters are HP critical before it emerges.
Once you locate and defeaat the hunt, return to the petitioner for your reward. They will give you the bounty, and the mark will then be closed and counted.
The marks are also set to classes between I and VIII. The higher the level of the mark, generally the harder it is to defeat. You should really try to get rid of as many of one level as you can, before you move to the next level. If you find you are having too much difficulty felling a target, level up some, and return. Once you accepted a mark, it will be yours for the remainder of teh game.
Also be careful when you choose a mark, and where they are. Some marks can hinder progress of certain areas, as well as prevent you from completing other hunts. A great example is "Battle on the Big Bridge". This hunt can be obtained fairly early in the game. But if you take it when it becomes first available, you will not be able to complete "The Child Snatcher" hunt, or explore the remaining parts of Lhushu Mines until the first half of the battle is complete.
Hunts are also a good way to open side events, complete Misc. Quests, or unlock new areas. There are also hunts that are required to be completed, in order to obtain certain Espers. So it is actually good to complete the hunts as they become available. Not only do you get a descent amount of LP, and fame in Clan Centurio, you also get some pretty darn good items, weapons, and armor.
Once a hunt is complete, you cannot defeat it again. Certain hunts carry great things that should be stolen. Make sure to steal from them, before they are defeated. You will not get another chance at them.
Once you become a member of Clan Centurio, Montblanc will also begin giving you hunts. These "Elite" marks are more difficult than teh ones found on the board, and usually yield far better rewards. To get these, simply speak to Montblanc, until he shows a list of available hunts. Then accept the hunt you want, just s you would at the board. In fact, every step of the Elite Marks is the same as the regular ones.