At Rosetta a number of troops unearthed a huge black granite tablet carved with the same message in three different languages: hieroglyphics, a cursive form of hieroglyphics called "demotic," and ancient Greek. A French translator, Jean Champollion, used his knowledge of both ancient Greek and the modern Coptic (Egyptian) language to translate the stone.
One important writing technique of the ancient Egyptians was the use of a cartouche, or an oval-shaped loop of rope drawn around the name of the pharaoh. In this case, the name of Pharaoh Ptolemy V frequently appears in cartouches. By plugging in the known letters he figured out, Champollion spent 20 years translating the Rosetta Stone and unlocking the meanings of the hieroglyphics. The Stone was actually a tax law passed by the Egyptian pharaoh to give priests a break on paying their taxes.
Naturally, there has been a great deal of work done by archaeologists and translators since the 1820s, but Jean Champollion's work with the Rosetta Stone made it possible to first decipher hieroglyphics. Being able to read the Egyptian writing made it possible to read a huge quantity of carvings, tomb paintings, papyrus scrolls, and great works of literature like the Book of the Dead. With all this material now available, modern historians (and students) are able to understand the culture, religion, and daily lives of the ancient Egyptians.
If you want to see the actual Rosetta Stone, you'll need to make a trip to the British Museum in London, or check out its website.