What is a "TV-L 13" position?
TV-L is the German public servant remuneration grade table (Tarifvertrag für den Öffentlichen Dienst der Länder (TV-L)). It is how civil servants Germany are graded for their salaries and similar conditions for their work.
Depending on where your position is, you'll be under TV-L West, or East, or Berlin, or Hessen. Something in your letter might specify this. Either way, there is information on the details at the Öffentlicher Dienst website.
Assuming you'd be in West Germany, taxed as a single (i.e., not married or living with a life partner or children), this boils down to a basic salary of €2103/month with a net salary after all taxes and health insurance payments of €1383/month.
This will increase as you remain hired, you'll go up the staircase of salaries, going from 1 to 2 after 1 year, from 2 to 3 after an additional 2 years, etc. Each step is a monthly salary increase of about €150/month net.
Found this link after Googling for "Tarifvertrag für den Öffentlichen Dienst der Lände" (Thanks to Mike for explaining first )
Tarifvertrag für den öffentlichen Dienst der Länder (Collective agreement for the public sector in the countries)
According to this, 13 is the pay group for Ph.D. students and postdoctoral associates. And the gross pay is from 3200 Euros upwards. (Basically it would be about 2000 Euros for 2/3 of that).
BTW, in one of the invitations, it says "E13". I'm not sure what this refers to but, I guess it's same as TV-L 13.
Hope this helps, and if you took this position, please explain further about TV-L 13.
Cheers....