I have a linux vps server and I want to see the current network connections and upload/download rate of each one. Is there any simple command that fulfill this purpose?
There are many tools available, although you will have to install a network monitoring system to get per-process bandwidth information.
In order of setup ease:
A simple package that does not require a lot of setup is iftop. This will show you bandwidth per process. This is probably what you want, if you want a simple real-time solution.