Madison argues in this essay that governments must, among their most pressing challenges, channel and control the activities and effects of "factions." Describe (including a few key quotes for each) 3 arguments he makes about how or why governments should do so.

In Federalist Number 10, Madison defines a faction as "...a number of citizens, whether amounting to a majority or a minority of the whole, who are united and actuated by some common impulse of passion, or of interest, adversed to the rights of other citizens, or to the permanent and aggregate interests of the community.”
Clearly, factions, as so defined, constitute a clear and present danger, representing as they do the triumph of sectional interests over those of society as a whole.
So, how should governments go about dealing with this implied threat? Madison makes two proposals. First, we should consider the possibility of removing the causes of factions. One way of doing this would be to restrict liberty. But immediately Madison steps back from the idea as its consequences would clearly be damaging:

But it could not be less folly to abolish liberty, which is essential to political life, because it nourishes faction, than it would be to wish the annihilation of air, which is essential to animal life, because it imparts to fire its destructive agency.

Air is one of the fundamental causes of fire, but we wouldn't try to reduce the danger of fire by restricting the air supply.
What else can be done? Madison considers another possible means of removing the causes of faction. This time, he considers the notion that we could create a society in which everyone has the same opinions and interests.
Aside from the practicalities involved, Madison's second proposal would've been unacceptably democratic to the Founding Fathers, giving too much power to the mob—the "swinish multitude" who lacked sufficient property to possess the disinterested perspective necessary to govern. Having floated the idea, Madison decisively rejects it:

As long as the reason of man continues fallible, and he is at liberty to exercise it, different opinions will be formed.

The problem of faction, then, cannot be controlled by removing its causes.
This leads Madison to conclude that the only way this can be done is to deal with the effects of faction. Why? Because,

The latent causes of faction are thus sown in the nature of man; and we see them everywhere brought into different degrees of activity, according to the different circumstances of civil society.

If faction is an expression of our very nature, then we can only really deal with its effects and try to minimize the harm it can cause.
To this end, Madison places much (some would say too much) faith in the republican nature of the Constitution itself. The system of checks and balances will act as a restraint on the domination of a minority attempting to subvert the interests of the majority:

If a faction consists of less than a majority, relief is supplied by the republican principle, which enables the majority to defeat its sinister views by regular vote. It may clog the administration, it may convulse the society; but it will be unable to execute and mask its violence under the forms of the Constitution.

But what about a faction of the majority? Madison, in common with the Founding Fathers in general, had a deep suspicion of democracy, fearing that it would lead to a levelling down of society, and of the affairs of the nation being placed in the hands of those without property—people who didn't have a stake in society and who would therefore govern purely in their own interests rather than the good of the country.
Representative government is the solution to this potential problem. This will, it is argued, lead to the election of men

whose wisdom may best discern the true interest of their country, and whose patriotism and love of justice will be least likely to sacrifice it to temporary or partial considerations. 

Under a republican system of government, men of property will take charge. As they are rooted to the land through their property holdings, and as they have sufficient leisure to consider the pressing matter of government, they are most suitably qualified to exercise power.
Madison also argues that having a large, as opposed to a small, republic will act as a check on the growth of faction. The larger the republic, the more "fit characters" there will be to choose from in the governance of the nation. Also the devious wiles of political rhetoric will be much less effective as there will be many more people to convince:

Extend the sphere, and you take in a greater variety of parties and interests; you make it less probable that a majority of the whole will have a common motive to invade the rights of other citizens; or if such a common motive exists, it will be more difficult for all who feel it to discover their own strength, and to act in unison with each other.