Yes, you should run a firewall.
Antivirus applications may stop rogue applications within your PC from doing damage, but external attacks don't require any action on your part.
You say PC, so I'll assume Windows. If you're running Linux, the specifics differ, but the generic issue is the same.
Windows runs services on TCP/IP ports that are exposed to the world. To connect to a printer, for example, it uses a network connection. And to publish its' name to a domain controller, it listens on a port.
It's possible for an external machine to attempt to connect to your PC on those ports. Most of those ports are now secured, and if you have a router, it's a lot more difficult for them to get to those ports. If you're behind an SPI (stateful packet inspection) router, it's extremely difficult. But if your PC is hooked up to the net via a DSL/cable modem, you could be visible to external attacks.
Go to https://www.grc.com/x/ne.dll?bh0bkyd2 and run their "Shields Up!" tool. It will simulate attacks on your machine and show you what was able to get through. It's harmless, but it will show you what exposures you have.
As for software firewalls, I've found that Armour Online's free version (www.tallemu.com) to be the best free firewall. Comodo and Outpost are also highly rated.