Composer is one of many Dependency managers. Composer makes it easier to distribute PHP code that relies on other code.
Installing it on your own host would look something like this
curl -sS https://getcomposer.org/installer | php
sudo mv composer.phar /usr/local/bin/composer
echo 'export PATH="$HOME/.composer/vendor/bin:$PATH"' >> ~/.bashrc
source ~/.bashrc
# now you can use it to do things like install the latest version of Drush
composer global require drush/drush:dev-master
while installing it on a shared host like WebFaction would be slightly modified
ln -s `which php54` ~/bin/php
curl -sS https://getcomposer.org/installer | php54
mv composer.phar ~/bin/composer
echo -e '\n# Composer\nalias composer="php54 $HOME/composer.phar"' >> ~/.bash_profile
source ~/.bash_profile
which composer
# now you can use it to do things like install the latest version of Drush
composer global require drush/drush:dev-master
drush --version
To use Composer in your PHP project, see the Composer Website, and also this handy-dandy (with mouseovers) cheatsheet