In the original game, the King periodically calls a vote to choose the new marshall. Two candidates will be listed and every other lord of that faction will cast one vote for one of the two candidates. If you are a vassal, you may cast one vote as well. The candidate you vote for will like you more (+2) and the one you didn't vote for will like you less (-2), regardless of which one wins.
If your Renown is high enough, you may be given the option to be a candidate, which you may refuse. If you are a candidate, the voting will progress in much the same way, except you don't get to cast a vote yourself. If you win the election, you become the new marshall and your opponent will like you less (-3).
The marshall isn't always elected, and sometimes the monarch will appoint a new one, especially if the current marshall has been defeated or captured.
Your standing with your faction or your friendship with particular lords seems to be of secondary importance. If your Renown is low, you may have a hard time being considered for Marshall even if your standing with the faction is 100. On the other hand, if you start off the game as a factionless bandit-hunter, rack up 1500+ Renown, then join a faction, you are almost guaranteed to get the title even if you've never spoken to any of the lords before.
If you are factionless and then accept a faction leader's offer of servitude, marshall elections very often happen soon after you take your vows (within a few game days or even hours).
Serving the Marshall
If you are not the marshall, you may occasionally receive orders from him when he starts a new military campaign. These orders are Quests. You do not have to do them, but if you accept them and subsequently fail any, you will lose reputation with the marshall.
The first order you will get is to report to the marshall with a minimum amount of soldiers. Typically the number of troops you will need will be much lower than you would normally have unless it is very early in the game. What types of soldiers you have doesn't matter; fresh recruits will work just as well as top tier knights or infantry.
Once you have joined the marshall, you will usually be told to follow him for a while. During this time, you must stay within sight of him. If you wander too far away, you will be warned to return. If you do not return quickly, you will fail the quest and irritate the marshall. You can automatically follow the marshall by right-clicking (or pressing square on PlayStation and X on Xbox) him and selecting 'accompany'.
The marshall may give you some other missions. These will supersede the instructions to follow him until you have completed them. Once the marshall's army has besieged an enemy fortification, the campaign will generally end and you can go back to your own business.
In With Fire & Sword, if the player is in one of the storyline quests, depending on the quest, even if the player is not the marshall, he/she will be able to give orders to lords of the same faction. However, it is not possible to call a campaign.
Being the Marshall
Being the marshall gives you the power to give orders and organize military campaigns. You will also have full authority over your allies in battle, so you may instruct their troops just as you would your own.
To give orders to your fellow lords (including the monarch), just speak with them and select "I have a new task for you." A list of orders will then be shown. To cancel orders, tell the lord "I won't need you for some time. You are free to do as you like."
If you wish to head a military campaign in Warband, talk to any vassal or companion and tell them "I want to start a new campaign. Let us assemble the army here." This will send out a message to all the vassals with instruction to meet you at that location.
You can also use any heroes you have in your party to start or end a campaign in Warband. Just talk to any of them, the option to start or end a campaign is the first one on the list.
In Warband, controversy can make your time as marshall very brief. Every time anything bad happens to your faction, your controversy can increase quite rapidly. The larger your faction, the more difficult it is to maintain order.
Once your controversy starts nearing 100, your monarch will want to replace you.
It is important to note that vassals will not always obey you. When given a direct order, if they are preoccupied with something else already (like heading to their castle), they may tell you they are too busy.
If they are not busy, they must listen to you, but may only do so for a limited period of time. This means that if you instruct a fellow lord to follow you and then besiege a castle, he may wander off before you even finish constructing the siege equipment. All lords, even ones that like you, will also be tempted to chase passing caravans or enemy farmers (this is less likely in Warband).
If you are not playing with realistic saving, you should save before starting the siege so that if all your allies run off, you can restore and try again. It is advisable to tell all your comrades to follow you again just before starting the siege, this will decrease the likelihood of them losing interest before the actual battle begins.
Calling together a military campaign can be even more frustrating, because it can take a long time for everyone to come together, or some lords may ignore your summons completely. If you only have one objective in mind, it may be faster to just manually give three or four lords direct orders to follow you rather than attempt starting a campaign.