What does an American recipe mean by “1 tablespoon vanilla”?
It means vanilla extract.

Whether it's correct or not is hard to say. It does sound like a lot for something with those quantities, so it's possible they meant to say a teaspoon, which is a pretty common amount, resulting in a subtle but noticeable flavor in a batch of chocolate chip cookies, for example.

Or it's possible they just wanted whatever it is you're making to really taste like vanilla, rather than just have a hint of vanilla in the background. Vanilla really is a subtle flavor, so it takes a lot to make it strong, and it's hard for it to really be overpowering.