Net salary for PhD in German university
Expanding a bit on my comment: To find out the net (and gross) salary for any position at a German university (from administrative assistants up to and including the base salary of a full professor), you can use the online calculator at Öffentlicher Dienst.

It's in German only, but not that difficult to navigate. You need to know the following information (I'm not going down every branch, just the one you asked about):


For temporary positions (PhD, postdoc etc.), select TV-L ("Tarifvertrag-Länder", wage agreement of the states);  for permanent positions (lecturers, full professors), select Beamte (civil servant).
The state the university is located in (since every state has full independence in matters of science and education; this mostly affects civil servant positions). In particular, you need to select whether it's a former West or East German state.
The year you are interested in (the wage agreement is renegotiated somewhat frequently, so it might change even within a year).
The pay scale (Entgeltgruppe); PhD and postdoc positions are usually paid according to E13; senior postdocs with more responsibility such as lab management can be paid according to E14. Civil servant positions can be either A13 to A16 for lecturers (depending on administrative responsibilities) or W1 to W3 for (junior to full) professors.
The pay grade (Stufe): For some scales, you move up a grade after several years (used to be by age, now it's by time on the job). This is something you have to negotiate based on previous experience; for a starting PhD student usually only Stufe 1 makes sense. (For TV-L 13, you get bumped up to Stufe 2 in your second year and to Stufe 3 in your fourth year; if you continue on to do a postdoc, you can get up to Stufe 5; see http://oeffentlicher-dienst.info/tv-l/west/stufen.html.)
The additional pension scheme (Zusatzversorgung). This is basically the organization that keeps the money deducted from your gross salary to put into a pension fund. If you're in one of the western states, choose VBL, otherwise VBL-Ost.
The working time (Arbeitszeit) in percentage of a full (forty-odd hour) working week -- this is the important one for TV-L. PhD positions can be announced as everything from 50% to 100%; postdocs usually as 75% or 100%.  Your 66% is about par for the course (depending on the discipline); positions announced as 100% usually include teaching duties beside research.
The additional salary (monatliche Zulagen) is important for W2 and up, but very unusual for TV-L. In your case, leave it empty.
The tax class (Steuerklasse) makes the biggest difference in your net salary. If you're single and don't have another job, choose class I. If not, things can get messy -- I'll just refer you to the German wikipedia entry and wish you luck.
The church tax (Kirchensteuer) or tithe again depends on the state; if you're not registered with one of the officially recognized churches, select keine. (Otherwise it's slightly less than a tenth of the income tax, which comes out as slightly less than 1% of the gross salary).
The health insurance (Krankenkasse) depends on the scheme you're in -- it's either one of the public insurance companies ("gesetzliche Krankenversicherung"), in which case you pay a fixed percentage of your salary (which varies slightly with the company; currently it's around 15.5%) or a private insurance company ("private Krankenversicherung"), in which case you negotiate a personal contract with a certain monthly fee. If in doubt, leave the default (public insurance, current average percentage).




Putting in the information you listed (66% TV-L E13, single, no kids) and guessing the rest (western university, current wage agreement, no church affiliation, default insurance), I get an average monthly net salary of €1497.36 (according to the current -- at the time of this edit -- collective agreement for 2016).