Do I need to install Laravel on a local computer to develop an application and later deploy it to a server with apache and PHP or the installation is on tbe web server where I can work on the development version before deploying it right there? It is not clear for me.
Everyone's develop/deploy setup is going to be a little different, but I think most here would recommend developing locally before deploying anything to a server.
That's is what I think as well. The question now will be, is there a deployment mechanism that packages everything together in order to upload or move all the needed files or I need to install Laravel on the server as I did on my local machine?
Research using git to manage your source code. With a git repository and a tool like Laravel Forge it's easy to deploy code to your server from your command line.
You should do whatever you prefer, there isn't a right or wrong way. As long as the development environment isn't production ;).
I develop locally using Laravel Homestead or Docker since a remote development environment isn't always available when working in teams.
Another benefit of developing locally is that the environment is always available, with or without an internet connection.
For maintaining your code base I would also recommend using git.
Can Forge also be used to deploy code to different environments @sutherland ? I thought Laravel Envoyer was the only "Laravel" tool for this job.