What can I do as OpenShift user?

I'm currently using a virtual server and want to try OpenShift out. But I'm not really getting yet, how it works. Do I get a root access to my "webspace"? Can I set up the server OS (e.g. Debian 7)? Can I install/uninstall software (nginx, PHP 5.5, PHP Code Sniffer PEAR package etc.)? Can I use one gear for multiple websites?

It not clear by your line of questioning what portion of OpenShift you are not understanding, so I will try and lay out the architecture and provide documentation to get you started.

OpenShift is a Red Hat developed product (so its going to be easiest to get started on RHEL or Fedora), but it can also run on other Linux systems (however you may need to piece meal the components together, but it can be done).

This is talked about in building your own live cd on the community site, however has not been done for you by the OpenShift community.

There are two starting places for OpenShift, and they dependon on what you are trying to use openshift for? As a PaaS hosting solution, or PaaS hosted solution?

For a PaaS hosting solution a good starting point is to look at the Origin page as it provides VM's and install instructions, for OpenShift's Community product.

Because OpenShift is a PaaS solution these components (see Architecture Links), when cobbled together provide users with an application space (which they do not have root access to).

https://www.openshift.com/products/architecture
https://www.openshift.com/wiki/architecture-overview

As the administrator for the box you would have (root access) but your end users would not.

For a PaaS hosted solution a good starting place for OpenShift is OpenShift Online which is Red Hat's Hosted solution for the OpenShift Origin project.

Get started by Creating an Acount

With an online account you can get started using the hosted solution very quickly by trying some of the quickstarts. Be sure to read the full set of OpenShift Documentation as well as install the client tools