Walk up to an NPC merchant and say "quest". If the merchant has a job they will ask if you will deliver a package to so and so in such and such a town. Accepting the quest drops a small crate in your main pack. You will need to take the crate to the indicated NPC. If you forget where you are going and who gets the package just left click on the package to be reminded. The package is labeled as a package for so and so in such and such a town.
If you want directions on finding the person, go to the town where they are and try to hand the box to any merchant. The merchant will say that so and so is in whatever direction. There is a 75% chance that you will deliver the package to someone in town. If your delivery is to someone in a different town you will be paid more for the delivery.
When you complete a merchant delivery quest, you now have a chance to increase your stats. The lower your stats are, the better your chances of an increase. Merchant delivery quests will pretty much only get you up to 75 or so in any given stat, though.
Walk up to a town guard or NPC Order/Chaos Guard and say quest. If the guard has a quest you will be told that such and such a creature is causing problems in random location. If you accept the quest you will need to go kill the creature, remove it's head and return it to the guard that gave you the quest. If you forget which guard gave you the quest, drop the head on any guard and they will point you in the right direction.
You can only have one guard quest at a time. If you try to get another one while you're still working on one, the guard will check to see if the quest you're doing is solvable (meaning that the guard who gave you the quest is still alive and that the monster you need to kill is still around somewhere.) As long as the quest is doable, they'll just remind you of the quest you're working on. If its not, they'll offer you a new quest.
When you return the head to the guard you will receive a reward. The reward you receive will be based on the number of quests you have done. The first quest in a series will usually result in a small cash reward. There is a slim chance that you could get a non-money reward on the first quest. As you do more quests the chance of getting a non-money reward grows. Once you receive a non-money reward the counter is reset and you will have a higher change of getting gold for the quests.
When you complete a guard quest, you now have a chance to increase your stats. The lower your stats are, the better your chances of an increase. Guard quests can increase your stats up to their maximum, and have a much better chance of an increase than merchant quests