Will I require a router for online play on a PS3?

May be...

I know I've asked a lot of questions lately about online but I'm new to this. It's a whole different world for me.

You do not need a router but it would probably be a good a idea to get a decent Wireless G router as im pretty sure your ps3 and your TV probably arent right next to your modem.

You can use a ethernet cord straight to a modem or a wireless router