Having router doesn’t mean you can’t be infected by mail attachment or by visiting malicious site – but I am guessing question is: is router sufficient for protecting PC from outside attack similarly as firewall does.
Answer is YES - under few assumptions:
You have disabled router’s “Remote management” in router interface (can be called sometimes as “WAM access” or similar depending on router’s brand and model).
If you want to be able to connect remotely to router you have made very strong user name and password and changed default router connection port.
All ports on your PC are closed (you don’t have in routers setting for instance enabled Universal Plug and Play (UnPnP)
Router will do same job as firewall, drop all not legitimate connection request coming from outside. Only thing which router (as oppose to firewall) is not able to do, is to control outbound requests coming from your PC to Internet potentially made by malwares present on your PC trying to call home. Still NAT routing will be probably less prone to misconfiguration errors than firewall. Unless you feel very knowledgeable and comfortable with handling security issues - ideal scenario: use both! (firewall and router)
PS.
If its wireless router, wireless security is separate issue which can be handled well by router as well as long as you using WPA (and not WEP for instance) for encryption and using EXTREMELY long and stron key.