Paris is a huge place that caters to a lot of different tastes, and there's an entire industry listing the best places to eat. Here I am just going to list some places that I have personally been to or that I have heard good things about from people I trust, plus some resources to find more.
Let's start with Jeu de Quilles, a great little restaurant with excellent organic wines. My wife and I had one of the best meals of our lives there in 2010 (thanks again Camille!). Very good ingredients - they're next door to the butcher shop that provides meat to the Palais Ã‰lysée. Very good quality for the price, meaning they're not cheap but the food is really, really good. (If you watch the 100th episode of No Reservations, you can see Anthony Bourdain eating there.)
Nearby there is Le Cornichon, recommended to me by the same friend who took us to Jeu de Quilles.
If you like sweet stuff, I have been told by Noah Falstein that Angelina is a must, although sadly I haven't been there myself yet.
I have never gone wrong following the advice of David Lebovitz, at least where food is concerned. David knows his stuff regarding chocolate and sweets, and he has lived in Paris for a long time. Paris Pastry Guide is an ebook and iPhone app listing his favorite pastry places in Paris.
Although it's a chain, I have a weakness for Boulangerie Paul, which can be found pretty much everywhere in Paris. I love their pain au raisins.
When I'm in Paris, I am often near Les Halles, and I like to eat at Le PÃ¨re Fouettard. It's not mind-blowing, but it's good French food.
I also like the CrÃªperie Beaubourg because it has decent crÃªpes and it's right next to the Centre Pompidou.
Opposite the Centre Pompidou is Amorino, a very good ice cream place. There's also one in Le Marais.
Completely by accident my wife and I had the best burger we've ever tasted in Les TÃªtes BrÃ»lées. Or, rather, she had the burger, and I had a mediocre salad. There's a lesson in there somewhere. Anyway, not something to go out of your way for, but boy that was a great burger.
After dinner I can recommend La Rhumerie. They have great rums, plus the best virgin mojito I've ever tasted. Once again, it was my wife's. She just has great taste.