If possible, try to start at Quartier latin, in the 5th and 6th Arrondissement. They are the best and most authentic areas in Paris, and close to everything interesting. Quartier latin is great for strolling at night, dining outside (weather and temperature permitting), and is close to the many sights of Paris.
Make sure you visit the quaysides with its second-hand bookstalls, and the Place de la Contrescarpe and Rue Mouffetard, where the nightlife is. There are many other possible places to stay, but if you can get a hotel here, you are located perfectly. Oh, and for a week, I'd choose a hotel, instead of an apartment.
Apartments usually rent out per month, not per day/week. Enjoy!
Actually there is no "best" arrondissement in Paris, Right bank has traditionally been associated with expensive restaurants and even more expensive hotels. Left bank brings with it bohemian images of University students at the Sorbonne and charmingly small and affordable hotels and bistros. Paris is divided into 20 arrondissement that begin at the city's center, Notre-Dame Cathedral, and circle clockwise to the North, East, South, and West of the city.
First timers should stay as close to city center as possible, although the 1st arrondissement is not always the best option as the hotel choices are limited. Some of the most popular options to consider are the 2nd: Opera, 4th: Hotel de Ville/Marais, 6th: Saint-Germain-des-Prés, and 7th: Eiffel Tower. But beware, within each arrondissement are smaller, micro-neighborhoods with unique personalities of their own which may suit you well or not at all.
And its important to find a hotel or apartment near a good metro stop. Paris is big. Very big.
And kind of like Las Vegas everything looks closer than it really is. You will either take the Metro often, spend a lot of money on taxis, or build up some really great stamina from walking. Do yourself a favor learn the Metro map.
And instead of choosing a hotel near your favorite museum or monument, find one close to a convenient metro station and you'll save both time and money. Staying in apartment would be cheaper option as most European country cost hotel per people not epr room. If you decide to stay in apartment, I personally recommending http://www.lodjee.com/paris, they apartment are near the center of attraction.
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