To begin with, please note that the Paris region is divided into 5 zones: zones 1 and 2 are core Paris, and many attractions are found in it, but of course not all. The more the zones you cross, the more your ticket prices increase. And public transportation in Paris is expensive, designed to exploit the huge tourist flow into the city. Options in form of public transport include metro and RER trains (the latter are longer distance, dirtier, and start and end outside of core zones), tramways (mainly in southern Paris), and bus lines. Some of the common options include a single t+ ticket (point to point, usually 1.7 euros), a carnet of 10 t+ tickets (13.7 euros total, so each ticket turns out to be cheaper), and Navigo/Navigo Découverte* passes**. The latter has to be bought for 5 euros by tourists; the former is for Paris residents and is immediately available for free of charge on producing proof of residence as well as a government-issued identity proof (usually passport for non-EU tourists). Once you have the Navigo pass, then you need to of course recharge it for the zones you wish to and for the time duration you want (week, month, or year). The passes give you right of unlimited travel in your selected zones during the chosen duration if you have a monthly or yearly pass. There is a dezoning during weekends, public holidays and certain school vacation periods, so that even if you have a zone 1-2 Navigo, you can still go till zone 5 with your pass without paying extra during all such periods as long as this falls under your overall selected duration. However, the dezoning does not happen for the weekly recharge of Navigo.
(A week starts on Monday, and the recharge for a new week can be done from Friday of the previous week till Thursday of the relevant week. A month starts on the 1st, and monthly recharge can be done from 20th of the previous month till 19th of the month in question.)
Now, many people buy a Navigo even though they are not going to realize its full value. If you are on a tight budget, saving 4-5 euros (or 9-10, if you are buying a new Navigo Decouverte card as well) also helps. When you buy a carnet of 10 tickets, then you pay 13.7 euros, much less than what you pay for a weekly Navigo valid in only zones 1 and 2 (more than 20 euros). However, if you are going to travel a lot in the week, then Navigo will be useful. If your stay exceeds 3 weeks or even if your stay is say 2 weeks but with a lot of going to all kinds of zones over the weekends, then a monthly pass (for which the Navigo is dezoned) - or if you are during those school vacations when dezoning takes place or public holidays fall in the week you are in Paris - might certainly be useful. If you are living for a longer time in Paris, then a mix of using both Navigo and carnet is the best option: you need to plan in advance, especially if you are a student in Paris.
Many tourists also buy a Paris Visite pass: if you are coming from one of the CDG or Orly airports, and if you have a Paris Visite pass covering all the 5 zones, then these will also be covered, so if you plan to stay for 3-5 days and visit a lot of Paris, then this is a good choice, as you won't have to pay extra for CDG/Orly access. However, if you are planning to travel extensively without counting the airports, then maybe you can buy Paris Visite for different zones or buy a Navigo instead.
Note that t+ tickets can be bought (singly/in carnet) and Navigo passes can be recharged at any metro/tram stop or RATP vendor. Buy your Navigo passes at bigger metro stations. Also note that a t+ ticket is valid inside all Paris (essentially zone 1), but not outside it (however, if with the interchanges on the same t+ ticket, you reach outside of a zone, then that's no issue: many metro lines end in zone 2). Note that you can take a tram/tram, tram/bus, bus/bus, metro/metro, metro/RER, and RER/RER interchanges with the same t+ ticket within 90 minutes (of course, while being in Paris). You cannot take a tram/metro, bus/metro, bus/RER, or tram/RER connection with the same t+ ticket. You cannot break your journey on the same line and continue with the same t+ ticket later on. You cannot return on the same line with the same t+ ticket.
*If you are a student in Paris and your age is between 12 and 25 years, then you can go for an Imagine R card instead of a Navigo, if you do plan to stay for at least 1 year. The card will come to you by mail in around 3 weeks, and you will require a French bank account to apply. The card will require you to compulsorily pay for 12 months' subscription, though you can of course prefer that to be divided into equal instalments.
**If your Navigo pass is lost, it can be replaced and any remaining credit in it at the time will be transferred to the new pass. In case of Navigo Découverte, however, you will need to buy it again (and any credit in the pass will be permanently lost - for you).
Prices and policies may change in the future.
Update, Dec. 2015/Nov. 2016: Certain Navigo passes now cover zones 1-5, so the point of Paris Visite pass is even more obsolete. If you have a Navigo pass covering zones 1-5 and if your destination is Orly, then you may well take Orlybus itself from Denfert Rochereau (and also a couple of stops later in south Paris). The bus usually takes around 30 minutes to Orly, but some days may see traffic jams.
