:repository is the Capistrano v2 version of specifying the remote repo you want to grab files from
:repo_url is the Capistrano v3 version of :repository, so depending on your version you'll use one or the other.
:local_repository is a variable you set if you want to grab files from a repo located on your own computer, or whichever computer you're developing on. So for :local_repository you would see something like
set :local_repository, "/home/eric/my_projs/proj1"
I found these answers at http://blog.huangzhimin.com/2013/11/02/upgrade-to-capistrano3/