This is a little confusing because the notion of what exactly "Adam" is is different from the original TV Series + End of Evangelion (Neon Genesis Evangelion) and the new movie series ("Rebuild" of Evangelion). In the original TV Series, Adam was a source of life (the Angels), and named the "First Angel", and used as a template of sorts for all of the Evangelions (eventhough near the end of the series, Evangelion Unit-01 was said to be a "clone" of Lilith).
In the new movies, "Adam" is a lot more vague. We're not sure what Adam is at this point, and the first thing that we have to go off of was a preview where we saw 4 glowing figures (presumably at the edge of the site of second impact) followed with the label "ADAMS":
Note that this shot was never actually used in the movies, it was probably reworked so that it more closely matched the rest of the Second Impact flashback montage:
Some things to note are that they all have halos and what appears to be "cores".
As far as the "Adam's Vessel", it's hard to decipher exactly what that means, especially if there is supposedly 4 Adams. However, we are told a few things from the movies, and that "Adam's Vessel" is Evangelion Mark.09. Here's a breakdown of possible "Adams" and what the 3rd movie has revealed.
It mentions that Mark.09 was referred to as "Adam's Vessel" and Evangelion 13 was called a "Survivor of the Adams" or something. The thread also goes into a little bit of detail and speculation about the Adams and what they could be. At this point, we don't know for sure what the Adam's Vessel was for, why it's called Adam's Vessel, and why Mark.09 is supposedly it.
More rampant speculation about Adams and the Evangelions here.