I have just two words to describe red Burgundy - Pinot Noir. Okay, I'll stretch it a little. Since Beaujolais is still considered part of Burgundy, I'll add another word that means a whole lot less - Gamay.
Let's start with Pinot Noir. Burgundy is the home of Pinot Noir, not California, Oregon, New Zealand or any other place under the sun. They've been making red Burgundy from Pinot Noir for well over a thousand years, and with a very few specific exemptions, red Burgundy is always 100% Pinot Noir. Something similar to Pinot Noir was described in the first century AD. The better wines are rich, complex and well balanced, with mild tannins, red and black cherry flavors and an underlying earthiness showing up as a mushroom or truffle note.
In the last thousand plus years, the Burgundians figured out which plots are superior to all others, and those (very few) vineyards are now called Grand Cru. There are more vineyards labeled Premier Cru - vineyards superior to all but the Grand Cru. In the next level of the Burgundy hierarchy are the village wines, followed by the regional wines, and finally by wines simply labeled Burgundy (Bourgogne).
Beaujolais is made from Gamay in the southernmost part of Burgundy. Beaujolais bears little resemblance to other red Burgundy, and probably should be kept completely separate, but it's not. Beaujolais is a lighter wine, less complex, yet still very good in its own right. Don't confuse real Beaujolais with Beaujolais Nouveau, which is meant to be drunk in the first few months after release. While not for long term aging, better Beaujolais will continue to improve for the first few years after release.