I am creating a Medieval empire. It is following the feudal system and I want to avoid the extreme political breakup that happened to Europe in the Middle Ages. How would I limit the power of feudal lords to retain an effective Monarchy?
According to historian Peter Heather, the force that leads to the weakening of central governments in feudal states of the Middle Ages is due to the limitations of the non-cash economy in these times.
Basically, in Ancient times, an Emperor could reward a follower with cash and jobs that come from tax revenues. These recur every year, so he can keep granting favors without losing his own assets and power.
In Feudal times, the grants must be in land, which the follower uses to support himself. A land grant once given, is gone, and the king is now poorer and the noble classes stronger. An exception is if the feudal state is on a border with a weaker foe, and can expand to grab more land or loot which can enrich a worthy follower but not eat into the King's lands. But this usually doesn't last long.
So the key seems to be for the King to find a "renewable" source of wealth or value that he can dispense to nobles as a substitute for grants of land.
Oldcat and SJuan76 have definitely provided a comprehensive answer to the OP's question, but I would offer another factor which might contribute to feudal fractioning.
The larger a kingdom grows, the harder it is for a King to stay informed about the challenges facing his Feudal lords. More and more, the lords are left to manage their lands alone or in concert with other local lords. Help when it does come down from the throne is often inappropriate in both timing and scale. It is only natural for the lords to begin to feel alone in their duties and to resent the meddling of the crown.
A charismatic King in close concert with all his lords, could easily maintain their allegiance by constantly highlighting the mutual benefits which arise from the bi-directional sharing of loyalty. But a King in absentia, isolated from current knowledge and unseen for long periods of time cannot inspire or maintain such loyalty.
So if you are building a medieval kingdom in a fantasy setting, and you want that kingdom to grow large and survive, give them a good communications network, maybe something magical, so that your King can stay present and involved in the lives of his lords.
-- edit to acknowledge a clarification from the comments --
In this way, a clever leader (King) can keep his supporting staff (Lords) from rebelling and keep an eye on any rebellious staff (Lords) so they can be replaced before they become a problem.
You could take Louis XIV's approach and keep the nobles so busy on attending the demands of the king that they don't have the time or opportunity to break away. I mean sure, the ultimate outcome of Versailles was the french revolution, but during Louis XIV's reign, it worked very effectivley to keep a tight grasp on his vassals.
Some possible approaches...
You could hold their families hostage. All feudal lords might be required to spend part of the year at the capital under the power of the King and leave their families behind when they go elsewhere. Japanese used to do this, IIRC. Similarly as CMaster pointed out the French used a softer variant of this where the nobility had to attend to the King at the court at the expense of having the time to build an independent power base elsewhere. Attending the Court was also expensive and made it harder for the lords to gather wealth they might use to challenge the King. A variant might require noble children to be educated at the capital.
Religion has been used to create cohesion. The Roman Emperor was also the pontifex maximus, the focus of the imperial cult, and later the de facto head of the church. The Chinese Emperor had the mandate of the Heavens to justify his rule. The Japanese Emperor was of Divine bloodline. And the Achaemenids were the protectors of the religions of their subjects and were in return largely supported by the various local religions. The point is that when the rule of the King is justified by religion, rebellion becomes evil thing to do and it is harder to get any support for it.
Alliances are very much part of the feudal system. All the noble families are vassals of the king, but some families have closer connections with the ruling family than others and generally less likely to rebel. Favor those families slightly to keep their support and make sure that all the critical locations are held by one of your close allies. Specifically, it should be impossible to threaten the capital without fighting your way through your most trustworthy allies. You also should prevent possible challengers from building their own alliance networks.
Fear is useful. You don't need to terrorize your subjects or vassals, but they should be aware that you can and will remove them, if they lose your trust. Basically you need to show that the fiefs are not permanent, if you have reason to decide otherwise. Even if that requires starting a war. Just remember to have that reason or your heir will have issues. And assassins do work.
Balancing factors to the power of nobility are good. Religious organizations (which should be controlled by or closely allied with the crown) can prevent nobility from having a power monopoly and support monarchy. In many countries the urban middle class was a key support for the King against the nobility. Cities might have walls and militias that give real military power. And the taxes they pay might allow a standing army loyal to the King to be stronger. A smart King might have recognized that the cities will become more important in the future and traded control over cities for giving nobility rights to the land. This would create a natural increase in the power of the crown over time. King should support the growth of cities and trade and protect the rights of the middle class.
Rule of Law supports the central government. So a strong legal system helps the monarchy. Provided the King is not obviously breaking the laws himself. If the legal system makes clear the the laws of the King are above the rulings of the noble Lords that makes the monarchy stronger.
Fall of Rome
You should keep in mind that feudalism is the result of the fall of Rome via barbaric invasions. So we should analyze what held Rome together and what feudalism lost when Rome fell.
A empire is a political entity based on two main factors:
The Roman empire grew steadily across hundreds of years. This means that during the timespan of its existence commerce routes and economic specialization were established. So, this leads to economic integration, meaning that a single political entity is unavoidable and something that can decrease wars and other forms of trade disrupting activities over a large surface. Another desirable consequence of a large political entity is capacity to build large infrastructure that foments commerce on a larger scale.
Romans were proud of roman identity. To be roman was to be civilized. This pride manifests itself as a desire to be part of the empire. There are no Romans if there is no roman empire. So this ideological concept is strong enough to bind the empire together.
When Rome fell due to the barbaric invasions both the trade routes, commercial specialization and national identity died, so in order to avoid feudal fragmentation (something tried by Charlemagne) you should fulfill those objectives.
Your empire should be built slowly, you should incentivize non-subsistence economical activity, so as to promote commerce. etc. keep and build routes. fund cities. etc.
One possibility would be to eliminate one or more of the top-tiers of nobility. In Europe this would be the Dukes.
This leaves you with the next step down - Counts - as the "top-tier" nobility. But because there are more of them, the individual power of any one Count is very reduced compared to the power of a single Duke.
Now, the King/Queen will need to create additional bureaucracy and military levels to handle what Dukes previously did, but they will have more control over those levels.
Of course, in total the Counts will have similar collective power to all of the hypothetical Dukes, but they won't be able to concentrate and apply that power as well. More importantly, the King can use his direct control to prevent the Counts from forming large alliances against him - he can use his levels of bureaucracy to break up large alliances.