The Paris train, metro, RER, tram and bus system uses a smart card called Passe Navigo Découverte for week passes. These passes are open to visitors (see note) as opposed to normal Navigo passes which are for residents of Ile-de-France only.
The Navigo Découverte pass can be purchased for a 5€ fee on top of the regular week pass prices noted above, which varies from 33,40€ for all give zones to 18,85€ for only zones 1 & 2 which covers only Paris. Five zones includes Paris and surrounding suburbs, airports Roissy-Charles de Gaulle (CDG) and Orly (ORY), Chateau Versailles, Fontainebleau, Parc Disney. The services included are any metro, RER (suburban) train, buses (RATP/Noctilien/Optile), and trams up to the zone limits of your pass.
The Navigo Découverte is sold at nearly any ticket window that normally sells tickets and passes for the Metro, RER, Ile-de-France trains. You’ll need a smaller than passport size photo to be used on the pass (exact Navigo Decouverte photo size is 3cm tall by 2.5cm wide), plus a pen to write your name. There’s nearly always a photo booth nearby ticket windows at major stations, costing 5€ for a set of four pictures, so be sure to get your photo taken before going to the ticket window where they’ll likely have scissors handy for trimming your freshly minted photo on the pass.
The downside of the pass is its timing. The pass validity starts on Monday morning, first service (usually around 05:30) and ends the following Sunday at last service (metro trains usually arrive at their terminus station around 01:40). If you happen to be arriving mid-week, it may not make financial sense to purchase the pass. After Wednesday midnight, you cannot add zones to the pass for the current week. From Friday onwards you can only purchase zones for the following week.
If you plan on arriving Thursday - Sunday, consider buying a carnet of tickets (book of ten or twenty) that are good for Paris zones 1 & 2, or day passes called Ticket Mobilis. Another option is to buy a Paris Visite card, which is really just a more expensive Mobilis ticket that can be used for a set period of days, starting from whenever you first place the ticket into a turnstile machine. The Paris Visite cards are good for 1, 2, 3 or 5 days and for either zones 1 to 3 or all five zones.
More information on the Navigo passes. Navigo passes can be used on the vehicles of the RATP, the SNCF (within the Transilien network), Optile, and companies under the aegis of the Syndicat des transports d'Île-de-France (STIF), as well as with the new Vélib' bicycle rental system.