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
This quest consists of guarding a merchant caravan to another city. The caravan has its own limited escort of Caravan Guards. Bandits will become slightly more aggressive towards you and your escort, making this type of mission an excellent source of combat experience. Note that you can only accept one escort contract (not including caravans you offered your services to directly).
You begin the quest with the caravan already trailing you. Speak to them once (required), then set out once prepared for travel.
Caravans move at a languid pace, easily allowing the player to scout ahead or hunt bandits that are attracted to the caravan but flee upon spotting the player's forces. The caravan's accompaniment is not sufficient to defend itself from a large band; try to keep it within line of sight at all times. Waiting is performed by default with.
Just before arrival at your destination, speak to the caravan. They will thank you for your assistance and compensate you as per contract. You do not and cannot take the caravan into the city itself, which may confuse unfamiliar players.
Simply take the wine/ale to the specified tavern and give it to the tavern keeper for your reward.
Hunt Down Troublesome Bandits
Upon accepting this quest, a party of Troublesome Bandits will spawn somewhere near the village. Simply find the bandits and eliminate them. Note: If an army attacks the bandits before you, you will fail the quest.
It's worth noting the bandits, or their equivalent(Sea Raiders), don't always spawn near the city where you get the quest. They sometimes don't even spawn in the zone supposedly attacked. If you have trouble finding them, you can ask caravans and nobles about them. In the case they did see them, they will tell you where and when, otherwise you will get a generic answer.
If you still have trouble finding said bandits, your best bet is to look for them in a zone where similar bandit parties spawn(Near Rivacheg for Sea Raiders, around Suno for forest bandits, in the mountainous area south or Praven for mountain bandits, near Ichamur for steppe bandits and near Shariz for desert bandits).
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.
Note that the quest must be finished completely within the time limit, including returning her to her family. If you save the girl and do not return in time, she will be stuck with you (likely a bug).
Make Sure Two Lords Do Not Object to Peace
For this quest, use your Persuasion skill to convince two warmongers (one from the town's faction and one from an 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 and will contribute to the completion of this quest, which is eliminating a total of around 4 Looter parties
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.