Does string theory have a notion of vacuum? Does string theory have a notion of vacuum? If yes, what is known about it?
 A: In the string theory literature, the term "vacuum" is usually synonymous with "perturbative string background", i.e. a target space of a 2d CFT with the right central charge.  This target space comes equipped with a metric satisfying the Einstein equations, as well as a host of other background fields, dilaton, B-field, and whatnot, all satisfying the equations implied by conformal invariance on the worldsheet.
Given any such background, you can use the machinery of perturbative string theory to add excitations.  A standard (but not terribly rigorous) argument indicates that adding these new  excitations is equivalent to deforming the background fields.  Which means that the new state with excitations is also a string background.  So any string theory state is a string background, and hence in some sense, a vacuum state.
It seems reasonable to think that any string background is a coherent state, built up by adding arbitrarily many strings to a state with no strings at all.  This hypothetical no-string state would be a better analogy to the usual sort of QFT vacuum.  But it's never been defined.  We don't understand what string theory is well enough to say if this true vacuum exists, or if all we can do is dial around through non-trivial backgrounds.
Similar comments apply to other corners of string theory, like matrix models and AdS/CFT.
A: As I recall from Susskind's course, there is no actual vacuum in string theory. There are some pieces of information, which can be helpful, like terminology developed for 2 decades.
Please, note the dates.
String theory is believed to have a huge number of vacua — the so-called string theory landscape.
Terminology starting from almost nothing:

"In discussing compactifications of string theory, we will discuss only vacuum states that can be described as the propagation of strings
  in a background space-time. It is quite conceivable that more complex,
  "inherently stringy" vacuum states should be considered, but a
  workable approach to considering them does not appear to exist at
  present."

Candelas, P., Horowitz, G. T., Strominger, A., and Witten, E.Vacuum congurations for superstrings. Nucl. Phys., B258, 46. 1985
Developed to more specific ideas:

"The vacuum structure of the theory, called the string theory landscape (or the anthropic portion of string theory vacua), is not
  well understood. String theory contains an infinite number of
  distinct meta-stable vacua, and perhaps 10520 of these or more
  correspond to a universe roughly similar to ours — with four
  dimensions, a high planck scale, gauge groups, and chiral fermions.
  Each of these corresponds to a different possible universe, with a
  different collection of particles and forces",

de Sitter Vacua in String Theory, by Shamit Kachru, Renata Kallosh, Andrei Linde, Sandip P. Trivedi,  2003.

"Flux compactications typically give very many possible vacua, since the  fluxes can take many difrent discrete values, and there is no
  known criterion for choosing among them. These vacua can be regarded
  as extrema of some potential, which describes the string theory
  landscape."

String Theory and M-Theory: A Modern Introduction, by Katrine Becker, Melanie Becker, 2007, p.477

"Gukov, Vafa and Witten (2001) made it evident that Flux compactications can lead to a solution of the moduli-space problem,
  since a nonvanishing potential for the moduli elds is generated. This
  led to the introduction of the string theory landscape, which
  describes a huge number of possible string theory vacua, in
  Susskind (2003)."
  
  
*
  
*Gukov, S., Vafa, C., and Witten, E. (2001). CFT's from Calabi{Yau    four-folds. Nucl. Phys., B584, 69. Erratum { ibid. B608, 477. E-print 
  hep-th/9906070.
  
*Susskind, L. (2003). The anthropic landscape of string theory.       E-print hep-th/0302219.

String Theory and M-Theory: A Modern Introduction, by Katrine Becker, Melanie Becker, 2007, p.715
Can't find any other notions dated after 2007. Hope this was helpful.
