Typically, shared hosting providers do not let you run your binaries on their host. In this case, for you to use Lisp they must have installed your Lisp implementation themselves and given you instructions on how to run it. This is also the case for Node, Ruby, Python — everything which does not fit into the model of CGI.
However, from the description on their site, Hostgator shared hosting is more flexible than others. They give you SSH access. In this case, the easiest scenario for you is to:
- configure a dedicated port number for your account (or request it from the support)
- copy your project to the hosting
- connect via SSH
- download your Lisp implementation
- run it at the configured port as you would do locally
for it to remain running after you disconnect, you must either
- detach it from the terminal
- maybe restas/contrib/restas-daemon.lisp
- or start-stop-daemon
- or just run it like "( sbcl --script my-launcher.lisp & )"
- or run it under a virtual terminal manager, such as "tmux" or "screen"
Unless you obtain a port number, you may combine Restas with FastCGI, or buy VPS, or learn how to use PaaS, such as Heroku or OpenShift. PaaS is straightforward when it comes to updating your application or ensuring that it's running, whereas other options give you choices and responsibility.