Brief Summary
The Set-Up
The Constitution's on its way, and people need to be on board with the drafters' ideas of what the Government should look like.
Hamilton, Jay, and Madison are tasked with publishing essays in the newspaper to get people on board with the new Federal Government. They need to get nine out of the thirteen states to support the Constitution, so a lot is riding on them being convincing as humanly possible.
The Text
While each Federalist paper was published anonymously, Federalist papers 10 and 51 were most likely written by James Madison, because they mostly deal with things about the government that he introduced. (Not so sly, JM.)
Federalist Paper 10 is all about warning the power of factions and competing interests over the United States Government.
Since everyone has their own self-interests, and people's self-interests clash with others', governments have to be able to pass laws for the common good instead of any one specific group.
To do that, the United States needs a Democratic Republic instead of a true Democracy, to cut down the power of the majority and filter it through (hopefully) qualified statesmen. This system is also made better by having a larger republic, which the United States hoped to be shortly.
Federalist Paper 51 proposes a government broken into three branches: Executive, Legislative, and Judicial.
Each branch should be self-sufficient, but each should have some kind of power over the other in order for them to keep each other from taking over the government. The Legislative branch needs to be split further into the House of Representatives and the Senate because it's the most powerful branch, and members of the Judicial branch need to be chosen by the President with the Senate's approval because they want qualified candidates for a position that lasts for life.
This style of government also helps keep down the power of factions, a recurring theme from Federalist 10.
TL;DR
A three-branch Democratic Republic will be able to preserve American liberties, while also stopping the Anti-Federalists from throwing rocks at our windows.