SSH
General
Secure Shell (ssh) is a commandline-tool for logging into a different computer over some network (e.g. the internet) and for executing commands on that machine, as if one would be sitting there instead of the own computer. So you use ssh to build a connection to the other computer and can then interact with it, using it's shell. It is commonly used to log-in to the frontend Nodes of a supercomputer.
Access
OpenSSH is the standard ssh client on Linux and Mac and it is freely available for everyone. On Windows you can use Putty or the GitBash (coming with ssh) which is also free.
Sometimes this access is restricted, so you can only connect, when you are within the university/facility and its network. To still access the Log-in Nodes externally, one can 'pretend to be inside the network' by using a Virtual Private Network (VPN).
Usage
Logging in with OpenSSH on a Linux Cluster is done with:
$ ssh -l <login> <cluster>
Here <login> is your user name and <cluster> is one of the frontend nodes of the system.
If you need to start graphical applications you need to enable X11 forwarding/X11 tunneling by your ssh client. For OpenSSH this is done by giving it the "-X" command line option:
$ ssh -X -l <login> <cluster>
which might or might not work depending on the your Operating System.
For more Security and easy of use you should consider setting up authentication via ssh keys.