Brief Summary
The Set-Up
Think of a treaty like an essay. It's basically an argument for telling two countries, which had been shooting at each other, to stop shooting. Just stop it with the gun stuff.
The intro to the Treaty of Paris establishes who's talking (the King of England and this newfangled United States thing that'll never catch on) and their sincere desire to be at peace.
The Text
In order to stop shooting, the differences between the two countries have to be resolved. They have to kiss and make up.
The most important "difference" in this case was that the United States wanted to be the United States and Britain wanted the United States to be Britain Lite.
But this treaty proved it: the United States was its own boss now.
The rest of the treaty goes after individual points. It does some important establishment of borders, travel rights, and guidelines on who gets to fish where. It sounds horribly dull now (wow: a treaty that states who gets to keep the lobsters in Maine vs. Prince Edward Island!), but in the late 18th century fisheries were a big deal. The U.S. of A. was basically a strip of land along a coastline—the ownership of crab pots for crab cakes was vital.
Lastly, King George's people actually wanted to do something nice for the folks who never gave up hope. The treaty specifically includes sections to keep the Loyalists from being unduly punished after the Revolution.
TL;DR
The United States is officially a thing now.