Summary of Federalist #10 and #51

Federalist #10
In brief: You can't eliminate the causes of faction without eliminating freedoms. Since factions will exist, we need to mitigate them. The constitution does so, both by its (federal and representative) institutions and by the larger polity that unification would create.
The Problem of Factions
Factions--a catch-all term for what we might call "special interests," "political parties," and pressure groups like the Moral Majority--worried Madison. If a faction grew large enough, it could impose its will on an entire nation, resulting in a tyranny of the majority.
Eliminating Factions
Madison argues that we cannot eliminate factions. To do so would require either denying civil liberties (worse than having factions) or enforcing conformity (impracticable). Factions are natural and the most common cause (according to Madison) is the unequal distribution of property.
However, even if we can't remove the causes of factionalism, we can mitigate the effects. Madison identifies both institutional and societal means of doing so.
Institutional Solutions
A representative form of government mitigates the effects of factionalism better than direct democracy would. Unlike the masses who reign under direct democracy, elected representatives are unlikely to be tempted by temporary passions because representatives must balance all the competing demands placed on them by various constituencies.
Furthermore, the Constitution's two-tiered (i.e. federal) government also helps mitigate this dilemma because state and local legislatures handle local issues, leaving only certain policy issues to the national government.
Societal Solutions
Besides these institutional solutions, the Constitution would also increase the size of the American republics by combining them into a single large polity. Larger republics have a greater number of interests and parties than smaller ones, diluting the effect of any one faction. Thus, a national union has advantages over smaller (state) republics, in that a faction in one state is unlikely to spread to other states. Importance Today
These arguments were reflected in the later debates about pluralism.

Federalist #51: "Men are not angels."
Madison continues the theme discussed in Federalist #47-50: The importance of checks and balances.
If men were angels, he writes, no government would be necessary. Because men are not angels, government is necessary--but this creates a paradox. We must give our representatives sufficient power to govern us, yet prevent them from using this power against us. After all, we cannot assume that our representatives will be any more angelic than the rest of us. Some have dubbed this "Madison's dilemma"
Madison sees the answer to this paradox in the Constitution's separation of powers. Each branch (judicial, legislative, and executive) has ways to limit (i.e. "check") the power of the other branches--likewise, each branch has ways to resist encroachments by the others. The self-interested ambitions held by members of Congress lead them to counter the ambitions of the president and the courts. "Ambition must be made to counteract ambition," Madison writes. Giving each branch a check on the other is central to the Constitution's strength; these balances and checks ensure that the constitutional structures will be self-enforcing.
Furthermore, this design divides those seeking power in two ways. First, it separates authority between states and the center (federalism). Second, it separates competencies within the national government (separate executive, legislative, and judicial branches). Not only can one branch of the federal check the others, but the states can check the central government.
Moreover, dividing up power into so many pieces helps mitigate the problem of factionalism (see Federalist #10). Factions would have trouble taking control of the entire government apparatus. This multiple division of authority "will render an unjust combination of a majority on the whole very improbable, if not impracticable." Seldom would the majority embrace "any other principles than those of justice and the general good." For example, the president with his veto, allied with the Senate, can prevent the House (seen at the time as the most dangerous branch) from taking over.