Vim has no concept of "project".
The closest you can get without installing clunky plugins is the "current directory": when you start Vim, the "current directory" is set to the directory where you started Vim.
In your shell, this is easy to manage:
$ cd /path/to/project
$ vim
:pwd --> /path/to/project
If you use gVim or MacVim, the "current directory" is usually set automatically to $HOME
if you start Vim without a file so, either you find a way to start Vim in an arbitrary directory or you use :cd /path/to/dir
as soon as possible.
Without argument, the :NERDTree*
commands open the NERD tree window in the "current directory".
You can use :NERDTreeToggle /path/to/dir
to make it display the content of a specific directory.
Or you can make sure you start Vim from your project's directory and NERD tree will do what you want it to do.