(2022) For __PhD Student__ : If your age is under 26 years old you're eligible to an ImagineR transport card. it's a one-year "cheap" contract that you can do as student (you will need your "attestation d'inscription", un RIB, etc.). You will pay only about 35 euro/month and SU reimburse 50% (described above).
For more details https://www.iledefrance-mobilites.fr/titres-et-tarifs/detail/forfait-imagine-r-scolaire  .. (Wafa)

(2021) For //everyone// : To apply for Navigo Pass Annuelle, and pay per month (It’s 75.20 Euro/month as of 20210924). For subscription, you will need to pay an extra 7.60 Euros for the first time.
 
First make sure you have all these:
  - Ordered List ItemOrdered List Item RIB
  - Justificatif domicile
  - An ID Photo (Just use the one you use to apply for Titre sejour is fine)
  - Passport/Visa

General rule of thumb to get things move fast, always be ready with all the documents. Print several
copies of them and give it in whenever anything seems to be in doubt
 
There are two ways you can do it. If you have everything at the earlier half of the month prior to the
start of the pass (e.g. You are ready by 15th of September to apply for a pass starting in October), then
you can choose to do it online. Pros, it's quite straightforward, you can choose to read in English; Cons,
you might need to wait to get your card.

  - Create your account (espace personnel) at : https://www.iledefrance-mobilites.fr/je-gere-ma-carte
  - Complete your profile with your ID photo, address etc.
  - Choose the Navigo Pass plan that is suitable for you. If you're old i.e. not eligible for the ImagineR, probably Navigo Pass annuelle is the one. Then choose the option to whether pay per month or in one go.
  - You will be prompted to provide a RIB.

//In-person Option//
Go to a train/metro station that provides the service, so far in my knowledge (my == Mafer), you can do that in Jussieu/Maison Blanche for weekdays, but on weekend Porte de Choisy is better. Bring along all the documents and the personnel will do it for you. Pros, you get your card straightaway, if you miss out anything they will tell you straight; Cons, if you can't speak French well, you need some luck to meet a nice staff (I met one!). You should be able to get your card on the spot.

(Mafer, with small edits by Sharbat)