"Denars" are also the currency of the real-life Republic of Macedonia, but the Mount&Blade currency probably came from "denarii", which were coins used in ancient Rome and are also the etymologic origin of Macedonia's denar. Many other countries also use the similarly named "dinars", "denier", "denari", "denario", and "denarius"; all also ultimately from the Roman denarius.
When you first start out, the amount of denars you possess varies depending on your choices during character creation. For example, a Hunter, Street Urchin, or Game Poacher starts with fewer denars than an Impoverished Noble, Page, or Squire.
If you are lucky enough, you may find a Lord who wants their taxes to be collected from a village, or a town (depending on what fiefs the lord owns). You should take these quests. First, go to the village, and start collecting taxes. The village relation to you will drop slightly and if you continue to take them (when it asks you if you want to), it will drop again and some villagers may attack you. The drop in relation with the village is very little and can be recovered in one quest for the village, so if you want them to like you, it is no problem. Once you have completed collecting their money, the quest will be marked 'completed' and tell you to return to the Lord who gave you the quest. This part of the quest, however, has no time limit. So you can just keep the money and spend it yourself. You can get about 2000-8000 from doing this, which can be especially helpful in the beginning of the game. The Lord won't be able to give you more quests if you don't complete it, but there are many Lords and one barely matters. Plus, it's easy to earn back those denars later on in the game.
To quickly earn money, winning Tournaments is ideal. Unfortunately, travelling to the tournaments can be long and boring and sometimes there are no tournaments being hosted at the time you ask a tournament master. You can make up to 4180 denars if you win the tournament with maximum bets on yourself, though repeatedly winning tournaments over a short span of time will decrease your maximum profits. Tournaments can be tough to win in the early game and if the tournament uses weapons that your character is not adept with, you may find yourself wasting money.
Another option is to loot a village. Goods (as opposed to equipment) sell for almost the same price as its base value, so when it says it sells for 500, it basically does (it depends on the town you sell it to). A village's loot usually sells for 4000-6000 denars. To loot a village, you must be at war with the faction the village belongs to.
Attacking a village, however, is dangerous and can ruin your reputation with your king if you are a vassal. It also reduces your honor by 3. If you already have a very negative reputation with the faction that owns the village, they will come and stop you. Since you can't stop looting, you will most likely be defeated in pitched battle. So it is best to loot when you have only just killed some farmers, and when you are not a vassal or mercenary.
While Lords coming to protect their villagers can be dangerous, if you choose to loot a village belonging to a poor lord, he may try to stop you with as few as thirty troops. If your army is strong enough, you can defeat him and take him prisoner, which would help your finances even more than looting the village did, assuming you can hold on to him until a ransom is offered.
Villages are slightly exploitable. If you attack by yourself and the peasants defeat you, you can attack again without waiting. Even if you are at 0 health. Since the fight takes no game time, you can try again and again as many times as you need. Each time you attack, you will lose 3 more Honor and the lord who owns the fief will hate you more and more. Other than that, there is no penalty.
Note: When you fight with 0 health, any hit on you will knock you unconscious - even if it deals 0 damage. So watch out for their stones. Blocking will still prevent death.
You can also use some merchanting strategies to gain a lot of denars. The best places to do this are Curaw and Dhirim which both sell iron for low prices. You will generally want at least 2000 denars to do this so you can buy enough iron. They usually sell iron anywhere from 70-150 when it is cheap. You should try to buy as much iron as you can before the iron hits above 150 denars each to maximize your profit as they sell for about 300 in certain Towns. Each time you buy one, the price increases so don't buy too many. Also, when you sell them, the price drops so the more you have, the less you make per bar. You can easily gain 2000-4000 denars from buying and selling iron. Salt is also an effective strategy, as buying it at Wercheg then selling it to nearby Rivacheg can sometimes get you a 250 denar profit. Ahmerrad in the desert regions (Warband) is also a viable place to try, as buying salt there and then selling it to the Vaegirs (or generally the other side of the map) can lead to a similar profit.
Merchanting is obviously benefitted by upgrading the trade skill, as buying prices are lower and selling prices are higher, but it also leads to a more effective "Observe the local prices" search (which tells you how much you may get buying certain goods at your location then selling them at particular towns), found in the marketplace dialogue.
Another idea to trade, especially in the beginning, is buying cattle in villages, slaughter them and sell their beef in the towns. Just go to the village center, speak with the Village Elder, tell him you want to buy cattle. You might be lucky, so they will sell it for less than 50 denars. Take five of them, slaughter them, so you will get 10 fresh beef. Now sell it at the town, you should get 700 denars by investing less than 250. So that is a little "starting boost", as you don't invest much and it doesn't take much time. However, cattle often cost much more, so this strategy is not always profitable.
Later in the game, whether you have joined a faction or are running your own, you will capture lords of enemy factions which can gain you a lot of money. A very good method for running an army of knights that is too large to sustain with food (morale drops too low), you can continually fight and win battles to constantly temporarily raise morale causing the knights to stay together. On top of that, you can constantly battle and defeat lords of other factions which will make your army happy and fill you pockets with money. If you have enterprises as well, you can be gaining money each week while you rampage through enemy territory defeating enemy lords.
Productive Enterprises added in Warband are the best way to make a large and secure income. A dyeworks costs 10000 denars but will make you roughly 500 denars each week. This will take 20 weeks to pay itself off, but since you must pay your troops each week, it will prove more valuable than you think. You can actually make money each week with enough enterprises even when your army is 150+ soldiers. Since they are in towns, they only fail to make money when the town is under siege or when an enemy faction owns the town. Villages on the other hand, will constantly get looted and your fiefs can fail to give you money at crucial times such as during war. Also, in order to protect castles and towns, you must have soldiers in them which often cost more than the castle or town's taxes. Also tax inefficiency comes into play once you have too many fiefs causing them to be less effective.
Negative denars is a glitch within the game that can be achieved by overspending on upgrading more than 1 of a unit type at once. For example, if you had 80 denars, and 5 vaegir archers ready to upgrade for 40 denars each, upgrading them all at once will put you at -120 denars. This can only be done if you have enough money to upgrade at least one of the units, as any less will make the game state that you do not have enough money. As the game was not designed to go negative, it will reset to 0 after entering a shop or spending/gaining money through interactions such as hiring or completing a quest that gives any cash reward, even if it is not greater than your defecit. This glitch is very useful, as you can upgrade units as much as you want, and remove your debts by recruiting from a local village and setting the counter back to 0