I'm planning to setup OpenVPN in my Ubuntu server.
Do I need to have 2 NICs, one for connecting to the internet and another for users to connect to the OpenVPN server?
No. OpenVPN can run fine with just a single interface.
However, in some setups, a second NIC can be handy.
If you want to have this:
ie, your OpenVPN machine is also a firewall, sitting between your internal network and the outside internet. In this case, a second ethernet card is almost required (I wouldn't recommend the hacks you could use to get away with just one card).
If you just want to have your OpenVPN server sitting on the internal network, with your router providing a portforward for whatever port you're listening on so that traffic can get into the OpenVPN server - no, you don't need a second network card.