Mount&Blade is completely non-linear; you can go wherever you want and do whatever you like with very few exceptions. All quests are randomly generated.
The quests will always be of the following types, but their specifics are somewhat random. You can cancel a quest at any time by speaking to the quest giver and saying that you cannot complete it, but you lose some reputation with them every time you do this, and companions who dislike failing quests will express their disapproval.
All quests have a time limit; however, once you complete the quest the time limit is disabled. Once you complete a quest speak to the quest giver to claim your reward. You can take on another quest straight after, but you can only have one quest from each quest giver active at a time.
Speak to a town's Guildmaster to receive these quests. He typically stands in a prominent location wearing finer clothes than the rest of the townsfolk.
Move Cattle Herd
Similar to the village equivalent. You earn some money and reputation with the town by doing this.
Escort Merchant Caravan
Another straightforward quest. There's increased bandit interest, but probably nothing you can't deal with. In fact acting as escort is an excellent way to get into battles more frequently. Note that you can only have one of this quest active at any one time (not counting caravans you offered your services to directly).
Because the caravan is so slow you can't easily lose sight of it, so you can go off and fight some nearby enemies if you want to (the caravan will also help draw them towards you if your party is still small enough that they don't run from you). Try not to run too far ahead of the caravan (zig-zagging or holding down Space will help). You will fail the quest if the caravan is destroyed, so if it is attacked be sure to run in and help.
Deliver Wine
Simply take the wine to the specified tavern and give it to the tavern keeper for your reward.
Hunt Down Troublesome Bandits
Simply find the bandits and eliminate them.
Ransom Girl from Bandits
This is a simple quest. Head to where the bandits are waiting for the ransom money. You can either pay them to release the girl or kill them and free the girl yourself. Attacking them before getting the girl will not endanger her life, so you only need to pay them if you are too weak to take them down. Note that if you fight them without paying the ransom you get to keep it, whereas if you pay them and then attack you won't get all the money back.
Once you have the girl, return to the town and she will thank you and leave the party. Talk to the Guildmaster to finish the quest and get your reward.
Make Sure Two Lords Do Not Object to Peace
For this quest, use your Persuasion skill to convince two warmongers (one from your faction and one from the enemy faction) to accept peace, bribe them to the same effect, or else ensure they are captured and can't interfere in the peace talks. If you are the marshal of your faction or the lord in question likes you enough, it is theoretically possible to go to your faction's lord, ask him to follow you, and then allow him to be attacked and defeated. It is also possible for this to happen without your involvement. Once both warmongers have agreed or been captured, peace will be declared. Return to the Guildmaster for your reward.
Deal with Looters
The Guildmaster wants you to hunt down Looters near the town. Accepting this quest will cause several Looter parties to spawn near the town. The quest parties have purple labels (refer to party details); normally-colored Looter parties will earn you normal loot and experience but will not contribute to the completion of this quest.
You are rewarded for killing the Looters and also for any goods you retrieve. Raising your Looting will cause Looters to drop more loot, thus increasing your profits. Once you destroy all of the specially-marked parties the quest will be complete.
Deal with Night Bandits
For this quest you will be attacked by bandits in a similar manner to the random attacks you may have already experienced. If you are successful you earn 200 EXP and 150 denars, and your reputation with the town increases by one point.