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.
Talk to the Village Elder to get these quests. He is the only villager not in motion, and generally stands in a prominent location with different clothing from what the rest of the serfs wear.
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.
Bring wheat
The Village Elder will ask you to bring some sacks of grain. Simply go to nearby towns and villages and buy the amount he asked for. This will earn you 400 EXP and +5 relations with the village. It will also raise their prosperity by four points.
Deliver Heads of Cattle
The Village Elder will ask you for a certain number of cattle. Go to the nearest village and buy the required amount of cattle from their Village Elder, then drive the cattle towards the village ("talk" to the cattle to start or stop their movement). When you get close enough the village will automatically take the cattle. Go inside and talk to the Village Elder to complete the quest.
Train the Peasants Against Bandits
This quest is fairly straightforward. After accepting it, a new item (Train peasants) will appear in the village menu. Clicking it will make time go by on the map until one or more peasants are ready for a sparring session, which you then have to win very much like in the Training Grounds. You will more likely than not have to repeat this process more than once to train the required number of peasants. Once you have trained enough of them, the bandits will attack.