rhubarb.vim

If fugitive.vim is the Git, rhubarb.vim is the Hub. Here's the full list of features:

Enables :Gbrowse from fugitive.vim to open GitHub URLs. (:Gbrowse currently has native support for this, but it is slated to be removed.)
In commit messages, GitHub issues, issue URLs, and collaborators can be omni-completed (<C-X><C-O>, see :help compl-omni). This makes inserting those Closes #123 remarks slightly easier than copying and pasting from the browser.
Sets up :Git to use hub if installed rather than git.

Installation

If you don't have a preferred installation method, I recommend installing pathogen.vim, and then simply copy and paste:

cd ~/.vim/bundle
git clone https://github.com/tpope/vim-rhubarb.git
vim -u NONE -c "helptags vim-rhubarb/doc" -c q

In addition to fugitive.vim, Curl is required (included with OS X).

Provide your GitHub credentials by adding them to your netrc:

echo 'machine api.github.com login <user> password <password>' >> ~/.netrc

If you'd rather not store your password in plain text, you can generate a personal access token and use that instead:

echo 'machine api.github.com login <token> password x-oauth-basic' \
  >> ~/.netrc

If you are using GitHub Enterprise, repeat those steps for each domain (omit the api. portion). You'll also need to tell Rhubarb the root URLs:

let g:github_enterprise_urls = ['https://example.com']

How do I turn off that preview window that shows the issue body?

set completeopt-=preview

Like rhubarb.vim? Follow the repository on GitHub. And if you're feeling especially charitable, follow tpope on Twitter and GitHub.