Paris is a great city to visit - great museums, amazing works of art in galleries, good food, interesting architecture and buildings. Unfortunately, this means lots of people do visit it, and with that comes problems.... As other questions mention, a lot of Parisians get fed up with tourists who make no attempt to speak French, and a small number try to take advantage of them :(
If you check WikiVoyage you'll see details of a few of the scams to avoid. For basically every one though, the key factor is they're only tried against tourists, especially tourists who don't speak French. Partly it's that if you spend much time there you'll learn to recognise the scam, and partly that you'll know how to report it to the police. As Mark mentions, they generally clear off when the Police appear...
As a slight aside, I had an interesting chat with my sister over Christmas about Paris. She learnt German at school, I learnt French, and our experiences of Paris were completely different. She'd suffered rude waiters and people trying to scam her, I've generally found it all fine. The difference seems to be a small amount of French.
For almost all of Paris, try and speak in a little bit of French first, as per this question. If it's a scam, they'll generally ignore you if they think you're a native, so a "non, merci" should be fine. If it's something real, they'll either switch to English, or ask someone else (eg someone stopping you to ask for directions). A bit of French almost always seems to help in restaurants, especially in tourist areas!
If you look and act like a local, you'll be fine, and you can have an amazing time in Paris. If you look like the baby animal that was separated from the herd and is now being surrounded by big cats, well, things aren't looking promising for you...
Quick summary - learn a few words of French, politely say no in French if approached on the street with something that might be a scam, and try not to look like a slightly confused+lost American tourist.
If possible, travel near a group who does look like that, the hunter pack will sense their weakness and take them down, leaving your part of the herd free to graze and wander ;-)
(Seriously though, I've been to Paris quite a bit, with levels of French varying from basic beginner through to my current levels, and the only time I ever had problems was when I was half asleep and responded in English to someone at the Sacré Coeur, and I got out of that one with one of the mentioned string bracelet thingies for the price of a coffee)