There are two kinds of people in this world: people who enjoy museums and people who don't.
Maybe this sounds pompous. Maybe it is. But if you don't see a reason for taking the time to look at works of art created dozens, scores, even hundreds of years ago, and beloved by other artists and critics and commoners for generations afterward, I don't think I can convince you to change your mind. Which is probably just as well: it means there will be less of a line for the rest of us!
Paris' museums are superb, so much so that it's worth planning your trip around their opening hours: there's nothing worse than paying 15 Euros for admission at 3 PM, only to learn that the doors close at 4! In most cases I wouldn't recommend more than one per day, unless they're small ones; you need time to process everything you've seen in order to benefit fully from the experience. Some of my favorites follow.
The Louvre: Perhaps the most famous museum in the world, it deserves all the notoriety it can rake in, the atrociously modern lobby notwithstanding. Stop here to see Nike on the prow of a ship, all the more regal for her headlessness; the ideal of the female form in the Venus de Milo; and Leonardo's La Giaconde.
Then ditch the crowds and explore some of the other collections: the Dutch Masters, especially its two Vermeers; the medieval foundations of the building, visible on the basement level; and the wealth of additional Greco-Roman sculpture that spills out into several light-filled atria. You could spend a whole week here, without sleeping, and still not see everything!
Afterwards, wander through the Tuilerie gardens, stopping to feed the birds (and yourself if you've been there long enough; there's a pleasant outdoor cafe about halfway through) and enter L'Orangerie, a small sun-filled outbuilding where you can stand in a room surrounded by dreamy, dwarfing Monets in murky pastel flavors.
Musee d'Orsay: Widely famed for its extensive collection of Impressionists, who were long banned from the artistic establishment, this former train station also houses Old Masters and architectural drawings. It's absolutely mandatory for fans of Van Gogh, Seurat and Degas; you could easily spend a whole day here.
Musee Cluny: This is my personal favorite. Housed in an ancient Roman bath, it is devoted to the art of the Middle Ages. It is home to the world-famous Unicorn Tapestries, beautiful stained glass, and countless illuminated manuscripts, which I think are the most fascinating pieces ever created. If it's a hot day, you'll be especially happy to visit the old frigidarium, which is buried to keep the temperature cool and consistent.
Musee Rodin: What I love the most about this place is its scale. You could see the whole thing, including the lovely gardens, in an hour, but you'll want to linger longer over each intimate piece, marveling at the softness of stone and bronze.
Centre Pompidou: I don't have much tolerance for modern art -- about an hour is the most I can take -- but this building is worth seeing just for its novelty: the idea was to put all the systems (circulation, water, heating and cooling) outside in order to leave the gallery space pristine and unencumbered. The plaza outside is also worth seeing: in the summer there are often impromptu concerts and skateboard performances, and the area is known for good shopping and restaurants.
Cite de la Musique was so wonderful I wrote a post about it several years ago, but this last trip introduced me to another wonderful gem: the Cinematheque Francaise, which houses hundreds of clips, props and costumes from the first films through the present day.
There are others, of course, but the last one I have to recommend is the Musee des Plans-Reliefs, an amazing find composed of immense relief maps created during the Napoleonic Era (all the better to conquer you with, mes amis.) It's only one long room, but it's worth the trip to Les Invalides even if you don't get to see the Little Man himself.
And sometimes you just get lucky, as we did the only time we ever got to the top of La Defense: there was an incredible exhibit at the Musee d'Informatique that covered the history of the Macintosh computer, complete with prototypes and projections of its most famous advertisements. Rob and I were in geek heaven, but we've never been able to go back since: high winds and technical difficulties keep preventing their elevators from ferrying tourists to the top.
Ah, well. It's Paris: there's plenty more to see.