$ composer install
If you really wanted to, you could do all the work that Composer does manually, but you definitely should not. Installing Composer is easy, it's just a matter of getting the
composer.phar file and running commands on it.
You do not need to run Composer on your server as well as locally, once you run
composer install or
composer update your project will have all it's dependencies available and you can just upload it straight to your server.