InDesign has far more typographic controls and other features for working with type, but it is a page layout application, first and formost. You can do simple illustration tasks using the pen tool and shapes, but ID lacks Illustrator's sophistication for illustration tasks, no brushes, no pattern fills, and no blends, for example. Neither ID nor AI has any abiltity to alter the pixels in a rater image other than applying some effects or blend modes.
Business cards are generally a very simple layout. As such they don't tax any program's abiliites very hard -- you can even do an passable job in Microsoft Word if you struggel a bit. If the task is going to require heavy illustration, Illustrator might be the better choice over ID since you probably don't need a lot of typographic sophisitication like multiple columns or anchored objects. If you need to manipulate a photo for the background, Photoshop is the place to be, and you can set type there, too, with limited capabilities.
On the other hand, if you plan on doing more projects, and they are going to be more complex or require multiple pages, InDesign is far and away the application of choice for most of us to do layout. If you are using art that's already existing and only need to design the page, not create the logo or color correct the photo, you won't need Illustrator or Photoshop, but like Bob I highly recommend getting the one of the suite packages so you have all the tools you'll need as you grow.