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.
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. There are two ways to acquire this. The first is you can go to a nearby village and purchase the required cattle. It might be best to visit a few towns before making a decision as prices can fluctuate wildly. Additionally, they also may not provide enough so visits to numerous villages may be required. The second option is to raid a village and steal the cattle, negating the need to pay for them. This will damage relations with the town that has been stolen from, as well as its nearby lord. Finally, be sure to herd the cattle, which has its own map marker, towards the town that requires the cattle, and the quest shall be completed.
Train the Peasants Against Bandits
This quest is fairly straightforward. After accepting it, a new item (Train peasants) will appear in the village menu (press tab). 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 (less than ten), the bandits will attack. Simply defeat the bandits who invade (both the peasants and your troops will help) and the quest will be complete. Be aware, if you lose the battle against bandits, the village will get 'looted' status and your reputation will not increase.