Yes. You get full root access to your VPS via SSH (short for "Secure Shell") and can manage the system independently. This allows you to install programs or analyze log files.
How to set the root password (Linux) or your administrator password for a Windows VPS is explained in these articles:
General information about which applications can be installed on Linux VPS can be found in this article: Which software can I install on my Linux VPS?
Please note, however, that by default we do not provide support for applications on your VPS and also do not have root access to customer systems.