Frames Per Second FPS?? How to tell 60 vs 30?

With the ushering in of the next gen consoles, everyone seems to be talking about the jump from 30FPS to 60FPS. I'm having a hard time discerning what this is exactly and I certainly can't identify a game running at 30FPS vs one running at a smooth 60FPS. I wish I knew how to tell the difference?

I know that The Last of Us Remastered runs at 60FPS so I'd like to be able to understand or appreciate this? Is there something that explains this? Is there a YouTube video that demonstrates what the difference is?